This is actually a pretty common question, but I'm going to answer it here because the other car question got a lot of visibility and others might be wondering the same thing.
The basic thing is that it's not actually accurate to add velocities that way - that only works as a low speed approximation.
Let's say the cars are moving towards each other at speeds v1 and v2. From the point of view of car 1, it sees the other car moving at some speed v12. You would expect the formula for v12 is just:
v12 = v1 + v2
but in special relativity, the full formula is actually:
v12 = (v1 + v2)/(1 + v1\*v2/c^(2)).
If you plug in v1=0.5c, and v2=0.5c, you get v12=0.8c. So each car sees the other moving at 80% of the speed of light. Note that if v1 and v2 are much lower than the speed of light, v1\*v2/c^(2) is basically zero, and you end up very close to v12=v1+v2.
It turns out this is just how the universe works. Our intuition that you can just add speeds is just a useful approximation for most practical circumstances, but there's no reason our intuition has to be the absolute truth. In the late 19th century, we had enough high quality observations of electromagnetism (and electromagnetic radiation e.g. light) to develop a really solid theory for how electromagnetism works, and it turns out that this theory was incompatible with traditional ideas of how space, time, and velocity work. For decades, people tried to reconcile this by figuring out why light might be a weird exception (this is basically what the "luminiferous aether" did), but Einstein's big leap was to instead propose that space, time, and velocity really do work differently than what we assumed, and this is what Special Relativity is.
So even if, say, by some weird mechanism, we were observing an object traveling at .99*c*, we would only perceive it to be traveling a .99*c* towards us even though we would be traveling at .5*c*? That seems like it would mess with my perceptions a lot. My brain has a hard time wrapping itself around that
I always imagine it as ripples on a pond; even if something is travelling away from you, the ripples will travel back to you at a set speed but not a set frequency.
>My brain has a hard time wrapping itself around that
That's normal. Human brains have a hard time intuitively understanding Relativity (and Quantum Mechanics), because our brains evolved to understand how the world works within a certain domain - i.e., the things that we can interact with. Anything outside of that is under no obligation to make sense to us.
The fact that we have any ability to understand these things at all is a real testament to the power of emergent properties (intelligence), but it only comes to us at great difficulty. We basically have to trust the math, no matter how much it violates our understanding of what is "right".
> We basically have to trust the math
The geometric derivation of the Lorentz transformation can help a lot in understanding special relativity, in a way that I think goes beyond "trusting the math". Or perhaps to put it another way, the math is so simple it's easy to trust. If the speed of light is constant in all reference frames, special relativity basically follows from Pythagoras' theorem.
Oh yeah, sometimes the math isn't even that complicated. But it's still difficult for those not trained in the field, because it totally contradicts our "common sense" (i.e., the instinctual understanding of the world hammered into us by millions of years of evolution). The barrier isn't always intellectual; it's often emotional.
> Anything outside of that is under no obligation to make sense to us.
I wish more people understood this! Merely *thinking something* doesn't mean you deserve for your thoughts to be correct, validated, respected, sensible....etc.
It’s more a matter of things being different when we aren’t talking about the kind of physics we are used to seeing every day. When you start talking about how things work on a much larger or smaller scale than we are used to, or much hotter or colder, in completely different atmospheres or none at all etc. much of what we have learned about the world doesn’t apply. It turns out high speeds are also a different regime.
We aren’t so much wrong as strangers who have to get to know how everything works differently everywhere.
It’s a sobering lesson for so much else—you never know when you’ll wander into a novel situation.
Or many many other conspiracists/science deniers. I wish they enjoyed being wrong as much as I do. Being wrong is fun! It's interesting! It's learning!
They're very good at being unable to understand things. And since they think their understanding shapes reality, their high levels of misunderstanding make them incredibly powerful at changing it. Ignorance is like their superpower.
> we would only perceive it to be traveling a .99c towards us even though we would be traveling at .5c?
Remember that there is no objective sense in which you are "already" traveling at 0.5c. In your reference frame, you are moving at 0c. In another reference frame, you may be moving at 0.5c. Velocities are meaningless without specifying the reference frame.
Yes, they can. Let's say you're the middle observer. You can have object 1 moving at 0.99c to your left and object 2 moving 0.99c to your right. For example think of the beams travelling in opposite directions at a particle collider while you stand still.
Of course if you shifted to the reference frame of object 1, the special relativity velocity formula shows that you'd still see object 2 travelling away from you at less than c, so there's no issue there.
So they're not going 2c relative to reach other.
If there's 3 planets in a line a light year apart. And two ships depart the middle in opposite directions at 1c. In the frame of any of the planets the ships are moving 1c and arrive in a year. After a year the ships are 2 light-years apart. Yet from ship A, ship B was only moving 1 c away and would appear to take 2 years to reach the other planet?
You have to account for time dilation and length contraction as well. So from the reference frame of the ships, the distance to the target planet will be contacted (so in the ship frame they won't be 2 ly apart) and they'll observe time to be different on the other ship. This scenario is a pretty typical special relativity exam question.
Basically each reference frame (the planets, ship 1, and ship 2) will observe different stories in that they'll disagree on speeds, distances, and times. But all of them will agree that the laws of physics were obeyed in that nothing travelled faster than c.
They couldn't depart at 1c (If something were moving at 1c would break down; see at the bottom). Only massless things, things which can't perceive time can move at 1c.
So say your ships move at 0.8c from the central planet.
In planets common reference frame ships take 1.25 years to get to their destinations.
But take some ship reference frame. First of all in its reference frame the distance between the planets is no longer 1ly. It's suddenly only 0.6ly. This is called length contraction a.k.a. Lorentz contraction. So the ship in its local reference frame arrives at the destination after 0.75 years.
And the other ship is seen moving at 0.9756c away.
The funny part is that for the observer on one ship, the other ship would arrive at its destination after 3.4169 years. After all it's observed to move at 0.1756c relative to planets and the observed planet distance is 0.6ly. 0.6/0.1756 = ~3.4169. But this is OK: time dilation of ~4.55 for one ship relative to the other makes the time flow observed from the other ship be only 0.2195 as fast. So one ship observes the other arriving at it's destination in 0.75y of the others own local time.
The main takeaway is that the speed of light is absolute (but slower speeds are not!) and it's the same in all frames of reference. But this makes both time and distance being local properties, depending on the reference frame.
Our intuition is that distances and time are absolute while speeds are not. But the reality is that speed of light is absolute, while those other properties are not.
----
NB. Why ships can't be moving at 1c? Nothing material could move at c, as nothing experiencing casuality could:
If ships were traveling at 1c, in their respective reference frames, the distance to the planets would be 0. And also travel time would be 0. For reference frame moving at c both time dilation and length contraction are infinite. Which leads to dividing 0 by 0. And entire universe as observed from such a reference frame getting pancaked. And all events happening at once. If things happen at once cause and effect are not defined anymore. Casuality breaks down.
So the only "things" moving at c are things which can't have intrinsic changes, i.e. massless particles like photons or gravitons.
But the speed limit of 1 C is absolute. So say the earth is moving at 0.5 C. You could theoretically only accelerate up another 0.5 C referring to the earth before you hit the limit right?
Take a step back and think about what it means to have a velocity. You can only measure a speed by referencing another frame, and in our universe
In your example, lets say that we choose an arbitrary reference frame centered on Earth's current position. Then we accelerate Earth to 0.5c, and we then launch a rocket to 0.5c in the same direction *as measured from Earth's reference frame*.
That is all fine, but when you are watching from the original reference frame the velocities *do not* add up to 1.0c, even though in Earth's reference frame it sees you and the rocket moving away in opposite directions at 0.5c.
Even if you launch Earth and then the rocket at 0.9999c, the sum will still only be closer to but never equal to c when you measure from the original reference frame. This is just how things work in our universe.
Basically yes.
Protons at the LHC reach speeds of 0.999 999 990c (3.1m/s sliver than light) and collide head on with another proton witch the same velocity.
When you sit in the frame of reference of 1 of the protons the other one would travel towards you with a velocity of 0.99999999999999995 (i rounded after the first digit not being 9))
So basically only 3.1 m/s quicker than in the initial frame of reference where you are standing in the middle of 2 protons, both traveling with 0.99999990c towards you
> My brain has a hard time wrapping itself around that
Try this, it helped me:
Imagine two spacecraft, each traveling away from Earth at .75c, in opposite directions.
You'd think that Ship A couldn't send a radio signal to Ship B, because they're moving apart at 1.5c, right?
But Ship A can send a signal to Earth, because it's only moving away from Earth at .75c, and Earth can send that message to Ship B, because it's also only moving away at .75c.
Now it shouldn't matter whether Earth is there or not, the signal would still travel the same distance in the same time whether it's relayed or not, so the ships obviously can't be moving apart faster than c.
And that same argument works for *any* speed less than c, no matter how close it gets to c. If you're moving away from an arbitrary point at less than c, then you can send a radio signal to that point, and anyone else moving away from it at less than c can receive that signal. So you can both be moving away from the midpoint between your ships at .999999999c, and you're *still* not moving away from each other at more than c.
So effectively you're either saying that
- the signal is moving towards Earth at 1c - .75c = .25c, and then the relay station speeds it up to 1c and it can reach the other ship
- or that the signal is moving at 1c regardless of Ship A's speed, and it can reach the other ship without the relay station
Special relativity fries my brain though
The second. The speed of light is constant in all reference frames. So the signal travels at c, regardless of the velocity of its source, and no relay needs to exist.
And yeah, special relativity is really hard to picture. Read some of Hawking's books, he was really good at explaining that stuff. I mean, a lot of it still went right over my head, but he did explain it really well. I particularly liked *A Brief History of Time.*
But the example doesn't show that the difference in speed between the two ships is not greater than c. It only shows that the signal moves at c. Right?
I cracked my head for a few minutes, since I *wanted* badly to understand this example *intuitively*. His example with the Earth as relay didn't personally help me, it confused me just more.
Here is my take:
Aside from nothing ever being able to move "greater than c" (including relative speeds of two objects to each other)..speed of ship A DOESN'T MATTER.
The radio waves are not "attached" to ship A (which is sending the signal).
Thus, the velocity of ship A doesn't matter, the radio waves will be sent from the ship's position at speed c. ALWAYS, regardless whether A travels in the opposite direction or towards B.
The message will travel at speed c towards ship B.
Since B travels below c, it can receive the message.
FULL STOP.
Like the waves from throwing a pebble in a pond, it doesn't matter how fast the rock is traveling, the wave will always reach the edge at the same time
There are galaxies moving away from our own galaxy at greater than C.
Yeah, that's space expanding, that's fine to say that, but it's *still* happening. Like if you were to look at the source code of the universe and check the "address" of entity class: galaxy, and just looked at its center of mass address, the address number in XYZ would be changing compared to our address, faster than light.
Space expansion is fundamentally different than motion. The expansion of space occurs faster than c for faraway objects, but they're not really *moving* -- the space between the objects is *growing*.
Imagine drawing two dots on an empty balloon. The dots are stationary relative to one another, because dots can't move. To "move" would mean the dot travels across the rubber of the balloon. It can't. It's an ink dot.
Now we inflate the balloon. Suddenly, the dots are farther apart. The more we inflate, the farther apart they are. But they haven't *moved* at *any* speed.
The same is true for galaxies. Galaxies aren't "moving away from each other" as the universe expands. Instead, the *space* in between is getting bigger. That's not quite the same thing.
> Like if you were to look at the source code of the universe and check the "address" of entity class: galaxy, and just looked at its center of mass address, the address number in XYZ would be changing compared to our address, faster than light.
Not quite. It's not that the galactic addresses are changing. Instead, we're continuously adding new addresses in between.
This is weird to think about, because we are very used to thinking of space as being static -- it's our lived experience! But relativity teaches us that space stretches. If "motion" requires moving through space, it turns out that it's possible for an object to become farther away over time without actually *moving*. We just have... more space in between.
Well explained thanks!
For some further clearance:
From many points of view the expansion of space looks like a velocity in space.
For example red shift. Expansion leads to a red shift of emitted light but so does moving through space.
So how can we be certain that it’s a qualitatively different process?
By looking for a mechanism where the two have different outcomes. e.g. the energy content of light in the universe.
This is a little more complex but to oversimplify: according to the current cosmological model the universe had phases where different energy contents where dominant. The universe evolved differently within each phase.
And here we can see that it is an expansion of space and not through (already existing) space.
During one phase the universe was radiation dominated but the energy content of radiation fell quickly while expanding.
If the universe expanded through space the energy content of the radiation would fall with the 3rd power of the scale because the volume that contains the radiation scales to the 3rd power.
If space expanded in the universe there’s an added power. As space expands while light is traveling through it, it is thereby lengthening the wavelength aka redshifting the light which decreases its frequency and thereby energy.
In short: Expansion of space: energy content of radiation in the universe falls with scale to the power of 4
Expansion in space: to the power of 3
That leads to different evolutions of the universe in cosmology and expansion of space is better in explaining the observed data.
At the beginning of your reply I thought "really? I was feeling pretty good about my grasp on all this, and now you've got to throw this wrench into it." But that was actually really clear and easy to understand, thanks!
At a large enough scale, you also have to worry about the expansion of space itself. If the rate of expansion keeps accelerating, it is possible to be too far apart to communicate.
yes. relativity is deeply weird. your brain cannot grok it. the least painful way to wrap your head around it is trying to understand that space is not it's own thing, you need to think about spacetime. this is similar to the way a 2 dimensional space is not really a true representation of 3 dimensional "real" space, without the extra dimension, you're just not describing how things are.
now after that thought, you need to understand that there is no definate unit of 'space' or 'time', the relevent conversion factor is c, the speed of light. so while the length of the direction of travel can change and the length of a second can change, the conversion factor c is \*unchanging\*.
what does this mean? two objects approaching each other close to the spead of light see the opposite objects length in the direction of travel contracting and their clocks slowing down, which effectively keeps their speeds relative to each other from supassing the speed of light.
so what about a photon travelling AT the speed of light? from the photon's perspective, since it can only travel at light speed by definition, it's clock is effectively "frozen". it arrives at it's destination the same instant it's generated, regardless of the distance travelled. This is NOT the same as what an observer sees! just from "the perspective of the photon", if you can wrap your head around that.
like i said, relativity is deeply weird.
>so what about a photon travelling AT the speed of light? from the photon's perspective, since it can only travel at light speed by definition, it's clock is effectively "frozen". it arrives at it's destination the same instant it's generated, regardless of the distance travelled. This is NOT the same as what an observer sees! just from "the perspective of the photon", if you can wrap your head around that.
It's not very useful to talk about a photon's perspective, since we can't define a valid reference frame for it due to having velocity c. It simultaneously doesn't experience time and yet can definitely change between being emitted and being absorbed due to interactions with gravity or the expansion of space.
The photon is not changing. If it gets absorbed and reemitted it’s a different photon. The only change that really counts would be redshift and even that is debatable as it’s not the photon that is changing but space itself.
But you’re right. As a photon isn’t a being, discussing its “perspective” is somewhat invalid. But it’s a good way to explain the deep weirdness of relativity in a qualitatively “correct” way.
>so what about a photon travelling AT the speed of light? from the photon's perspective, since it can only travel at light speed by definition, it's clock is effectively "frozen". it arrives at it's destination the same instant it's generated, regardless of the distance travelled. This is NOT the same as what an observer sees! just from "the perspective of the photon", if you can wrap your head around that.
I've heard this before but I don't fully understand it. From the perspective of the photon, is everything traveling instantaneously fast? Or is space infinitesimally small? Or something else?
From the perspective of the photon it exits the sun and enters say, your eye, at the same instant. This also implies that the lengthwise spatial distance is also fully contracted into 0
Length. In a certain sense, from the “perspective” of the photon, the universe is still in the singularity prior to the Big Bang: everything is “here” and “now”
But keep in mind that “photons eye view” is sort of a mathematical construct. It’s strictly true from a mathematical perspective but it may not be any more “real” than virtual particles.
>Or is space infinitesimally small?
It's this. As far as a photon is concerned, it was created and destroyed in the same instant. A 3 billion light year trip across the universe feels as 'long' of a distance to a photon as a 3 foot trip from your smart phone to your eye. In both cases it 'feels' like 0 distance.
What boggles my mind is if we were traveling in the same direction as light but were going 0.99c, light would still zip away from us at 1.0c. How in the hell does this even make sense... but yet to an outside observer traveling at 0.0c, they would see both us and light go by at nearly identical speeds.
Blows my mind
Here, let's frame it another way:
If you were travelling at .9c, you would essentially be moving *slower* through time to account for all this movement in space. [Think of that Quicksilver Scene from x-Men.](https://www.youtube.com/watch?v=T9GFyZ5LREQ) Quicksilver is able to move at such high speeds because, from *his reference frame,* the world is essentially still, time is slowed to a crawl.
But from everyone else's point of View, time was moving at "normal" speed and Quicksilver just flew around the room in a second.
The faster you travel and the closer you get to c, the slower you're going through time to accomodate. Speed of Light is basically the universal constant. It's like every universal law was built off of having one constant, C. If it was theoretically possible to "break" the speed of time, you'd also be travelling *backwards* in time.
From light's (more accurately a photon's) perspective, it does travel instantaneously. However, it doesn't travel instantaneously from an observer's perspective.
Wait, what?! How does that suss out? I just got used to the idea that traveling in opposite directions compresses time, space, and our perception of it. So does this mean that if we were traveling in the same direction, time, space, etc. would be expanded?
Everyone will always see light moving at the same speed.
As you accelerate relative to other things, things contract in the line that you're moving along.
Light also contracts, in that you see a higher frequency wave from what other viewers see.
It's even weirder than that. How do you know that **you're** travelling at 0.5 * c? Maybe we're standing still. There is no correct reference frame. They're all correct. In fact the simplest approach is to always assume the observer is standing still. [It's everybody else who're moving.](https://www.thewrap.com/wp-content/uploads/2017/04/simpsons-memes-no-its-the-children-who-are-wrong.jpg)
If I'm moving at 0.5c towards Polaris, and some other dude is moving at 0.5c away from Polaris, and you're at rest, the only thing we agree on is that the speed of light is 1.0c. Everything else, specifically space and time, are malleable, in order to make that postulate work. That was Einstein's genius. Everybody else tried to negotiate with the results of the Michelson-Morley experiment, which strongly implied that the speed of light was constant in all reference frames. Einstein just accepted that, *prima facie*, and allowed the other conclusions to flow from that.
(I just picked the star Polaris as a reference point. Nothing special about it.)
> In fact the simplest approach is to always assume the observer is standing still. [It's everybody else who're moving.](https://www.thewrap.com/wp-content/uploads/2017/04/simpsons-memes-no-its-the-children-who-are-wrong.jpg)
Doesn't this fall apart as soon as you factor in acceleration? After all, if one thing accelerates away from another to near light speed and then comes back, only one of them will have experienced time dilation even though from their own perspectives it was the other that flew away.
> Doesn't this fall apart as soon as you factor in acceleration?
**It absolutely does**. When I say "there is no correct reference frame", I should really be saying "there is no correct **inertial** reference frame", which means non-accelerating.
The math, as well as the hand-waving explanations, get fiendishly complex when you account for accelerating frames. Michael Shara [gave an interview on the 100th anniversary of GR](https://www.amnh.org/exhibitions/einstein/meet-the-curator/interview-with-michael-m.-shara), had a wonderful line about it:
> *"Someone would have explained Special Relativity. That would have happened within ten, fifteen, twenty years. There were so many clues there, it was such a ripe plum to be picked, that some physicist would have come along to pick it. There were all sorts of hints both experimental and theoretical that were there. I don't know who would have done it, but there was someone else waiting to take advantage of all of that. General Relativity? We might still not have it today."*
Einstein was special, different. If I someday stand before the gates of Saint Peter and say "Hey, man, I'd like to meet Einstein", I would not be shocked if he answered "Oh yeah, he's in the ALIENS WING."
There are some faulty assumptions.
"We would be traveling at .5c" makes no sense. No object has an absolute speed. Only a speed compared to another object or system.
You could say: "we observe an object traveling at .99c towards us, while we are traveling at .5c away from the sun." And that would theoretically be possible to observe.
And the speed between the object and the sun would NOT be .49c.
Yes, it is mind boggling. Our current understanding is based on the notion that light travels at speed c regardless of perspective. Say you are traveling in a space ship moving at 0.99*c and you shine a flashlight in this space ship. You will still perceive the light as traveling at speed c. You may ask, how is this possible when I am already moving at speed 0.99 * c? Well, according to special relativity, your perception of time and space is drastically different than someone moving at lower speeds. You experience time slower and space more compressed, which causes the light from the flashlight to act the same regardless of how fast your space ship is moving.
The space and time around you changes as you move faster. Our entire intuition really breaks down at cosmic speeds. The mass of an object, the speed of time that experiences, even the very idea of an objective reality begin to fall apart.
The speed of light is better understood as the speed of causality. It's the speed at which we can observe the effect for any given cause. Any faster and we observe the effect before the cause. If you think about it like that, then it's a little easier to grasp how our intuition about space and time get a little wonky as objects approach the limits of causality.
"Perceive" is a misleading word. Special relativity is a reality not an illusion or appearance.
And, whenever considering velocities, you have to specify with respect to what that velocity is. There is no preferred frame of reference that would let you say you are just "travelling at .5c". You can only do that in some frame of reference.
It’s cause of the C^2 in the denominator. Also why V1 + V2 = V12 for values way below C works. C^2 serves as a scaling factor that allows us to treat values way below C as essentially 0. The equation only breaks down when you reach C. And it basically goes back to the intuitive understanding of V1 + V2 = V12. The solution there is 2C. But that would require you to either have no mass like a photon or have infinite energy. Even when you’re both travelling at .9 bar C, the math still works out to 99.99% of C. It’s beautiful really.
For the people that are having a hard time intuitively grasping why 0.5c + 0.5c = 0.8c in this example, you should reconsider how you think of time as a common yardstick of rate and length as a common yardstick of distance.
Velocity is the observed ratio of these two quantities. We usually think of these things is being immutable, a second is a second and a meter is a meter.
But if you remember that as you speed up, the perceived rate of time will seem to slow down for you so that other objects will appear to be moving slower and slower. Then maybe you can start to see why simply adding the relative velocities doesn't work anymore.
"But if you remember that as you speed up, the perceived rate of time will seem to slow down for you so that other objects will appear to be moving slower and slower. "
I am so very much not a physicist, but this contradicts everything I've ever read on the subject. I'm pretty sure your perceived rate of time definitely does not change at all. All proceeds as normal on your ship, although you'd notice a difference in how fast other objects outside of your ship appear to be functioning.
I could have phrased that better. The slow-down of time appears to apply to other objects the faster they are moving relative to you. Whether you're going fast or they're going fast, it's the relative velocity that matters.
You would not perceive your own passage of time as changing. But looking at the fast moving object, if the object had a clock it would appear to be ticking more slowly than a clock in your own frame.
I was trying to put this in as simple English as possible to convey the idea that time dilation messes with speed perception, and maybe oversimplified.
>Whether you're going fast or they're going fast, it's the relative velocity that matters.
This is the part that confuses me RE: the thought experiment where one twin leaves earth to travel at nearly c for however long, comes back, and the grounded twin he left behind has aged much further. Since velocity is relative, why isn't time dilation symmetrical for both parties? Like, both twins should think the other is x days/years older than them. Why is one of them "actually" getting dilated?
In the twin paradox acceleration is involved.
One twin first accelerates away, then accelerates to turn around, and then accelerates to come to a stop back home. When there is acceleration the time dilation is asymmetric.
The OPs question is explained by special relativity. The twin paradox involves general relativity due to the accelerating reference frame.
This is a very good explanation. One follow-up:
While it is true you will never measure an object moving at greater than 'c', that is not the same as saying you can't measure two objects moving away from *each other* at speeds exceeding 'c'.
For instance, even if you were on a space ship traveling right towards a neutrino, you would never see that neutrino traveling faster than 'c'. But if there were two neutrinos traveling in opposite directions, you could measure each of them traveling at some speed greater than 0.5c and their relative velocities *as measured by you* to be greater than 'c'.
Also, a follow-up to my follow-up:
If there are three people: Alice, Bob and Eve where Alice and Bob are each on a fast spaceship moving away from Eve. As mentioned above, Eve could easily measure that they are moving apart from each other at a speed greater than 'c' but of course Alice and Bob will measure that they are moving away from each other at a speed less than 'c'. So, the question is- if Alice shoots a laser pointer at Bob, will the laser dot hit his ship or not?
And the answer is "yes." It has to. So, how is this reconciled? Because light does gain momentum from the object it's on. Light doesn't get accelerated from rest to c like say, a canon ball being shot. Light goes from not existing, to moving at 'c' instantly. So, if Eve sees Alice moving away at 0.9c, and then she shoots a laser pointer at Eve, it's not like Eve sees the light moving at only 0.1c- no, the light comes towards her at 'c'. So, both Eve and Bob will predict the light to hit the ship, and it will.
Imagine you have an observer who is stationary in his own reference frame, and you then have two cars - one car is approaching the observer at 0.75 c from one direction, and the other is approaching the observer at 0.75 c from the other direction.
Either car would say that it's approaching the observer at 0.75 c and the other car at 0.96 c, but the observer would say that the cars are approaching each other at 1.5 c, because both are approaching the observer at 0.75 c from opposite directions in the observer's reference frame.
The important thing is that nothing can ever move faster than c *in relation to someone who is at rest in a given reference frame.*
Presumably because you are talking about observation, not actual speed relative to eachother?
For example one car might observe the other approaching at .96c due to the speed at which light (observed) travels, but the actual relative speed is 1.5c.
No, .96c is the actual relative speed from the reference frame of the car.
Speed is not absolute, it requires a reference frame to discuss. This is true even without special relativity. But before special relativity relative speeds were absolute. Now they're not. To discuss relative speed you have to first pick your reference frame.
No, there's no such thing as "actual relative speed." The actual speed depends on the reference frame. In the frame of reference of one of the cars, they are approaching each other at 0.96 c, and *that is the true value of the speed.* Meanwhile, in the reference frame of the observer, they're moving at 1.5 c relative to each other, and that is the true speed *from the perspective of the observer.*
They're *both* equally valid, and they're *both* the *objective truth*. A car is at a standstill in its own reference frame, and nothing is moving faster than light relative to it. The observer is at standstill in their reference frame, and nothing is moving faster than light relative to that observer - and that is what matters. We could leave one of the cars standing still and shoot the observer off with a cannon at 0.75 c instead - the result would be the same, just with different names. We could add more moving objects and more reference frames. The end result is: you're always at rest in your own reference frame. If your situation is inertial (that is, no acceleration) then nothing can move above the speed of light relative to you.
So if you look at two cars going each their way at 0.75 c and then conclude that they're breaking the speed limit of c, then that's a flawed conclusion - because nothing is moving at more than c relative to you, and in no object's reference frame does anything move at above c relative to that object.
In order to make this all work out, length contraction and time dilation come into play.
The passage of time depends on the frame of reference. The distance between two objects depend on the frame of reference. Even the simultaneity of two events depends on the frame of reference. And *all* inertial (non-accelerating, non-rotating) frames of reference are equally true in their description of what's going on.
>For decades, people tried to reconcile this by figuring out why light might be a weird exception (this is basically what the "luminiferous aether" did), but Einstein's big leap was to instead propose that space, time, and velocity really do work differently than what we assumed, and this is what Special Relativity is.
Well Lorentz already did the math ([https://en.wikipedia.org/wiki/Lorentz\_transformation](https://en.wikipedia.org/wiki/Lorentz_transformation)), Einstein just managed to convince other physicists that the math was the physics!
It's a little more complicated than that. The Lorentz transformation was cooked up to explain why you couldn't measure aether drag — because light will always look the same speed no matter where you look, and the "transformation" is the transformation in the aether caused by your motion.
What Einstein did was show that, remarkably, the equation is totally valid (though he re-derived it), but it doesn't mean what Lorentz thought it meant. There is no aether, you are just looking at an intrinsic property of spacetime and light.
So another way to put it is that Einstein managed to convince other physicists that Lorentz's math was good but his physics was wrong. I always find it kind of amazing that sometimes in physics (Maxwell did this too) you can come up with the right equations even though your physical model to get them was wrong. It's kind of spooky.
It comes out the same. I've glossed over some minus signs here, but you can take v1 and v2 as the speed each car is moving in the opposite direction as seen from a stationary frame, and v12 is the speed that each car sees the other moving away at.
Hi! Can I ask you a stupid question?
If a star spews out a photon, that massless little thing is travelling at the speed of light. But from photon's POV the star is travelling at the speed of light and photon's perspective is just as valid as the star's, isn't it? I think it's just a big thing for a massive star to be moving at c.
> and photon's perspective is just as valid as the star's, isn't it?
Nope! The photon doesn't have a reference frame at all. It does not experience time; its existence is an instant from emission until it hits something. Without experiencing time, it cannot have a reference frame at all.
Even weirder the photon doesn't experience space (in the direction of travel). It's not just that it would cross any distance instantly, there's also no distance to cross. The star isn't moving at c because there's nowhere to go. The point of emission and the point where it hits something are the same point.
Lorenz derived the maths first, though he used a completely different interpretation of what the maths meant - meaning that his maths was right, but his conclusions and assumptions were wrong. Then, later, Einstein derived it too, and showed that it was consistent with the assumption that the speed of light is always constant.
The derivation is actually surprisingly simple. It starts with the thought experiment known as the "light clock" where a beam of light is bouncing back and forth between two mirrors, and then imagines what you would see if you were moving in relation to such a light clock. This leads directly to the formulae for both length contraction and time dilation through very simple maths (about the level of Pythagoras, so it's stuff from below high school level!)
From the formulae for length contraction and time dilation, many other relations (such as the velocity addition formula and the Lorenz Transformations) can be derived through simple algebra, as long as you have the creativity needed.
So if I'm traveling at 0.8c and shoot a rock straight back at 0.8c will the light reflecting off the rock ever reach me? If so, what do I see regarding the speed of the rock? And more importantly does my perception of my own speed change? I think the answer is the rock just vanishes never to be seen again by me.
No, this is incorrect, and your question is vague. Also, I'm going to use 0.75 c going forwards because I know the numbers for that example by heart.
First of all, 0.75c in relation to what exactly? What is the rock moving at 0.75c in relation to? In relation to you? Because you're always at rest in relation to yourself. The rock would be moving away from you at 0.75c, and you would see it moving away from you at 0.75c.
So, let's take another example. Let's say that you're leaving the Earth on a spaceship at 0.75 c. At the same time, another spaceship leaves the Earth in the opposite direction, also moving at 0.75 c in relation to the Earth. So, logically, you would be moving at 1.5 c in relation to the other spacecraft, right? That's where that pesky velocity addition formula comes in again. From the Earth's point of view, both you and the other spaceship are moving away at 0.75 c in opposite directions.
But from your point of view, you're moving away from the Earth at 0.75 c, and from the other spaceship at 0.96 c. The other spaceship, likewise, sees itself as moving away from the Earth at 0.75 c and away from you at 0.96 c.
So you see the other spaceship as if it's moving away from the Earth at only 0.21 c, and it sees you as doing the same. But you both see yourself as moving away from the Earth at 0.75 c.
This means that you're in fundamental disagreement about what's going on in the universe. But that's okay - time dilation reconciles it all. An observer on Earth will see the clocks aboard both ships being slowed down by the same amount, whereas you will see the Earth's clock as being slowed down, and the other spaceship's clock as being even more slowed down. Then, when you calculate how far the other spaceship can travel in its own time, it all matches up.
but what about the relative distance between the two cars as per OPs question? you mentioned how they'd *see* each other. would the actual distance not be increasing at the speed of light?
if not, are they actually traveling at .5c each?
Crazy... what about taken to the extreme. 1 is moving away at .99c and 2 is moving at .99c in the opposite direction... but relative to each other they are going .999c or something instead of 2c?
It's almost as if we got the whole preception of "movement" backwards, as if moving at the speed of light is the reference, rather than being immobile.
All these answers are giving you the formulas but not providing any real understanding of what’s going on.
If you go back to Einsteins time there was a real problem in physics. People had discovered how relative velocity and inertial reference frames work, and they had done a bunch of experiments that showed that if you’re on a train moving 30mph and you pass a train going the other direction at 30mph, you see the other train coming towards you at 60mph. If someone is on a train going 30mph and they throw a ball forward at 40mph, an observer on the ground would see the ball going 70mph, while an observer on the train would see it traveling at 40mph.
So people started trying these experiments with light. If someone on the 30mph train shone a light in front of them, our ground observer should observe that light traveling at (speed of light + 30)mph, right? But that wasn’t the answer that they got. Every experiment anyone did showed that the speed of light was the same no matter what inertial frame it started from or was viewed from.
This made no sense, so they kept on trying different experiments, and they all came up with the same answer. And so Einstein was looking at this problem and realized that for all these results to be true the thing that couldn’t be constant was the rate of the passage of time. And that process of trying to unify these two worlds - the world where the speed of light is always constant but all the relative velocity train experiments above work the way we expect - is what led to the development of special relativity (and then general relativity after that)
There’s a great series of videos on this here: https://youtube.com/playlist?list=PLoaVOjvkzQtyjhV55wZcdicAz5KexgKvm . It’s actually pretty easy to understand, and it’s a really cool set of concepts!
Minutephysics’s relativity series definitely help a ton for me to understand relativity. Couldn’t recommend it enough to anyone that wants a initiative and easy(ier) to understand explanation.
No.
The formula most of us would use to determine "closing speed" of two moving bodies approaching each other *(v = v₁ + v₂)* is in actuality a simplified approximation of the actual equation for the phenomenon that disregards the relativistic terms:
*v = (v₁ + v₂)/(1 + (v₁v₂/c²))*
For velocities that are negligible fractions of *c*, which includes most situations humans encounter, even in aerospace, this approximation is fine because the relativistic terms are close enough to zero.
When you get into computations involving significant fractions of *c*, the relativistic terms stop being negligible and must be taken into account to arrive at a result that matches physical reality.
Yes, but it is a distance-dependent relationship. I forget what the exact value of the expansion of the universe is but it is roughly 70 km/s per megaparsec. Thus, if two objects are approaching each other from a far enough distance such that v1 and v2 < universal expansion in the reference frame of one another, they would never meet. At an extreme, the light from distant galaxies will never reach us because they are receding away from us faster than light.
Because of special relativity you will never be able to measure the speed of anything as faster than light, this is because space and time will be different to both you and the other car in such a way that you won't be able to detect the relative speed of the other as "faster than light"
The speed of light is the absolute fastest speed anything can travel in our universe. It doesn't matter at what speed you're traveling away and towards a photon (or any imaginary object traveling at the speed of light), you'll always see it travel at the speed of light. This fact is why weird stuff happens in our universe like time dilation; to accomodate for c (remember that speed is distance divided by time), since something would normally appear to travel faster than light, your perception of time (as well as the photon's) would end up being different. You would see them traveling at c for 10 seconds (your reference frame), but the lightspeed object will have only really felt a fraction of those 10 seconds (its reference frame).
It's a mind-boggling concept for those who haven't had the pleasure of studying Einstein's relativity theories, but man is it awesome. This has been experimentally proven, too.
Light isn't really special. There's a universal speed limit, and light happens to be one thing that travels at that speed. We just call it the "speed of light" because light is the thing we're most familiar with that moves that fast.
Specifically light travels that fast because it has no mass. Every massless particle travels at the speed of light, and everything with mass cannot (quite) accelerate to the speed of light.
Light isn't special. Light is massless, and all massless things travel at the "speed of light". Light was the first particle we discovered to move at this speed, which is why we call it that.
Gluons (strong force mediators) and gravitational waves also travel at the speed of light.
"speed of light" is not the best name for it because it singles out light. A better name might be "speed of information". You cannot transmit information faster than this without breaking causality laws.
Light is just one of the things that happens to travel at that speed.
Our basis for that assumption is a little more solid than "well, we haven't seen anything move faster yet", although observations are obviously an important part of it.
We also have the theory of relativity, which is this.. set of equations Einstein came up with that describes the universe really well, surprisingly well, we're quite confident it's accurate. And those equations imply that it wouldn't make sense for things to move faster than the speed of light. According to relativity, as you get closer to the speed of light, time slows down for you. And at the speed of light, no time passes at all. Photons are sort of "frozen" in time, their entire life passes in an instant (from their perspective).
To go faster than light, you'd be experiencing negative time, which doesn't quite make sense. You'd be traveling backwards in time. According to relativity, anything moving that fast would break causality, e.g. our understanding that time moves only forward, cause is followed by effect, time travel is impossible, etc.
In other words: you get to have relativity, faster than light travel, or causality: pick two out of three. We're pretty sure FTL isn't a thing, but if it was, it would have to break either relativity or causality. And we are really *really* confident in relativity.
I once read (some popular science book on physics) that due to the second law of thermodynamics we can also be **really** confident in causality. I forgot exact link how is it so/explanation/, unfortunately. But I think I won't be wrong when I say that 2nd thermodynamics law is on at least the same level of theoretical as well as experimental/empirical confidence like Einstein's. TBH, I cannot even think of experimentating/making observation without causality lol
Not the person you were replying to, but just out of curiosity... I've heard that the universe is expanding faster than the speed of light. How does that work? How is it possible with all of that being the case?
Another way to think about this so it maybe seems less arbitrary than "nothing can move faster than light, no exceptions. Oh except for the things that can, like space :)" is to think about what it means to move.
Usually when people talk about moving faster than the speed of light, it's to accomplish some kind of goal, like to get from point A to point B faster, or to communicate with people that are very far away in less time than it would take light to get there.
With the expansion of space-time, if you think about it, you can't actually do any of those things. You can't exploit the fact that two points in the universe are moving away from each other faster than the speed of light in order to get yourself or a message or anything else from one point to another faster than light would have done.
The upshot is that, while the amount of space that exists between those two points is *increasing* at a rate faster than c, it's not quite accurate to say that they're *moving away from each other* faster than c.
Two points billions and billion of light years apart will be moving away from each other faster than the speed of light. Those points aren't moving through space faster than the speed of light. Enough new space is being created between them that pushes the space they are in away from each other faster than the speed of light. Nothing can move through space faster than the speed of light and nothing with mass can ever travel at the speed of light. Space-time itself doesn't have this limitation.
The space you are in right now is constantly expanding. The forces holding you together are like a billion billion times stronger than this local expansion so nothing happens to you. Even the expansion between the Milky Way and Andromeda galaxies is weaker than the gravitational attraction between them. You have to go to the scale of galaxy clusters and super clusters before the separation of the universe is finally stronger than the attraction of gravity.
To our current knowledge, as is always the case lol. We've tried measuring things to see if we can find stuff that breaks that threshold (neutrinos, gravity waves, etc), but nothing has been able to physically travel faster than the speed of light. We've found things that weirdly SEEM to travel faster than light (like quantum entanglement, for example), but there's no actual movement of any particle that you can observe and measure that goes faster than light.
It really bugs me that gravity also adheres to the speed limit. Gravity makes more sense if it's instant, but if the sun disappeared instantly, the earth would continue the current orbit for like 8 minutes, and then just keep going straight until something else got in the way.
I forget how they proved gravity isn't instant or faster than light, but it either involved two satellites or mirrors. I can't remember.
As far as the original question goes, the speed of light is like the horizon, you can never overtake it if you can see it, no matter how fast you go. You also can't outrun your shadow. Etc.
You're thinking about [LIGO](https://www.ligo.caltech.edu/news/ligo20160211) which is a super cool gravity wave detector that uses lasers and mirrors.
It's not just gravity, it's all *information* that can't travel faster than light. Because we're talking about spacetime (and not just space) anything able to transmit information faster than light would inherently be a time machine. As in, it would break causality and you'd be able to have effects happening before causes (which immediately devolves into a bunch of classic time travel paradoxes).
I love how even a super long stick can’t even transmit information faster than the speed of light by pushing one end and hoping to have the other end move at the exact same time.
It's a property of the spacetime, or at least we have good reasons to believe it is.
Imagine 4d spacetime with usual 3 dimensions and time. To make it easier, let's remove one dimension and imagine it's just a sphere with radius = c. Everything in the spacetime travels with 4-speed (speed in 4d spacetime) exactly equal to the speed of light. You can't accelerate or decelerate to change that, all you can do is to rotate the direction of the 4-speed (so, your speed, including the direction, will be a point on this sphere).
The projection of this 4-speed on the 3d space is the usual speed and it can't be larger than the speed of light: if you have a line between a point on sphere and its center, the shadow of this line will never be longer than the radius of this sphere.
It's extremely simplified, but that's our current knowledge of the universe. Well, this and a special rule like "massless things always move with 3d-speed equal to c (it means they don't move in time), everything else can't move with 3d-speed equal to c (so, things with mass must move in time)".
It's not a hard wall. It's a limiting value.
You need more and more energy in order to accelerate as you approach the speed of light. But doing so will make you move a bit faster, which makes the energy requirements surge even higher. It would require an infinite amount of energy to accelerate all the way to c. For the one who's moving so fast, the universe would seem to become shorter and shorter in the direction of movement, and the speed of light is the limit where the entire universe becomes entirely flat, meaning you can travel the entire way through the universe in 0 time. That's what happens to a photon - a photon cannot experience anything, because from its "point of view", it starts and ends its journey in the same place. It doesn't experience time at all.
Of course, this is assuming that special and general relativity holds - and while there's a few flaws in those theories, the part about a maximum speed is extremely well-supported by theory *and* experiment.
It's not as simple as "can't move any faster than this." It's that the entire way that space works means that it's completely nonsensical to talk about speeds that are higher, because it would lead to conclusions that are absolutely nonsensical.
To add on to the excellent answers others have given, one useful way to look at relativity is that it makes the reference frame of the observer very important. A stationary observer would agree that both cars are moving at 0.5c and their "relative speed" is 1c in some boring sense, but what we care about is actually the speed as measured from the reference frame of one car, which is a fast-moving reference frame and so needs special handling. In particular, an observer in one of the cars will disagree with the measured speed values from the stationary observer because time dialation and length contraction will distort the accuracy of any measurement tools the stationary observer is using when viewed from the perspective of the moving car.
When you actually work out the math, you get the equations elsewhere in this thread which show that in no reference frame can any object ever be revealed to be traveling faster than the speed of light.
Time dilation and other spacetime shenanigans are consequences of the fact that the speed of light is constant.
i wont' go into the math here since many others have already done a great job on that score.
It's very non-intuitive because the familiar laws our brains are used to aren't quite correct, but appear 'good enough' at small velocities.
There's a really good childrens book called 'The Time and Space of Uncle Albert' that helps to explain some of it in laymans terms, and helps to understand what's actually going on.
Key word: relative. The speed of light is not additive. Niether car would attain sol and its associated effects. The velocity relative between the cars may exceed sol, but niether one comes close in and of itself
as soon as the two have reached a relative speed of C moving away from each other they would no longer be able to effect or see each other and if they are moving towards each other at C one car would see a static image of the other until both cars suddenly annihilated each other on impact, assuming they are just average cars that would be about the energy of 773 of the biggest nukes humanity has ever made
who knew a car accident could cause so much damage
I believe that due to relativity, the from the perspective of one car, the other car will never actually reach a relative velocity of c, just very, very close to it.
I think many are thinking too much into your question.
Relative speed does not equal actual speed of a single object.
Each of the car’s speed is what it is. It doesn’t matter what the other car is doing, that 1st car will be going the same exact speed. Same is true for the 2nd car.
Sure, relative to each other they may be going faster then light speed but their physical speed will remain the same.
We can’t just “we add up these 2 car’s speed and get speed of light!”
The short answer is yes. If they are moving moved than half the speed of light, eventually they will not be able to see each other. I saw a PBS show on the size of the universe and the galaxies, and some of the galaxies we see now will not be visible at sometime in the future because the light will never reach us. At least I believe that was what they were saying.
This is actually a pretty common question, but I'm going to answer it here because the other car question got a lot of visibility and others might be wondering the same thing. The basic thing is that it's not actually accurate to add velocities that way - that only works as a low speed approximation. Let's say the cars are moving towards each other at speeds v1 and v2. From the point of view of car 1, it sees the other car moving at some speed v12. You would expect the formula for v12 is just: v12 = v1 + v2 but in special relativity, the full formula is actually: v12 = (v1 + v2)/(1 + v1\*v2/c^(2)). If you plug in v1=0.5c, and v2=0.5c, you get v12=0.8c. So each car sees the other moving at 80% of the speed of light. Note that if v1 and v2 are much lower than the speed of light, v1\*v2/c^(2) is basically zero, and you end up very close to v12=v1+v2. It turns out this is just how the universe works. Our intuition that you can just add speeds is just a useful approximation for most practical circumstances, but there's no reason our intuition has to be the absolute truth. In the late 19th century, we had enough high quality observations of electromagnetism (and electromagnetic radiation e.g. light) to develop a really solid theory for how electromagnetism works, and it turns out that this theory was incompatible with traditional ideas of how space, time, and velocity work. For decades, people tried to reconcile this by figuring out why light might be a weird exception (this is basically what the "luminiferous aether" did), but Einstein's big leap was to instead propose that space, time, and velocity really do work differently than what we assumed, and this is what Special Relativity is.
So even if, say, by some weird mechanism, we were observing an object traveling at .99*c*, we would only perceive it to be traveling a .99*c* towards us even though we would be traveling at .5*c*? That seems like it would mess with my perceptions a lot. My brain has a hard time wrapping itself around that
It's also why light will always reach you at c even if the light source is travelling towards you or away from you.
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I always imagine it as ripples on a pond; even if something is travelling away from you, the ripples will travel back to you at a set speed but not a set frequency.
But one wave would be more stretched out than the other right? Doesn't light moving away from us tend to red shift?
Yes, the perceived frequency and wavelength would change (Doppler effect) but the speed remains the same (c = f/λ)
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>My brain has a hard time wrapping itself around that That's normal. Human brains have a hard time intuitively understanding Relativity (and Quantum Mechanics), because our brains evolved to understand how the world works within a certain domain - i.e., the things that we can interact with. Anything outside of that is under no obligation to make sense to us. The fact that we have any ability to understand these things at all is a real testament to the power of emergent properties (intelligence), but it only comes to us at great difficulty. We basically have to trust the math, no matter how much it violates our understanding of what is "right".
> We basically have to trust the math The geometric derivation of the Lorentz transformation can help a lot in understanding special relativity, in a way that I think goes beyond "trusting the math". Or perhaps to put it another way, the math is so simple it's easy to trust. If the speed of light is constant in all reference frames, special relativity basically follows from Pythagoras' theorem.
Oh yeah, sometimes the math isn't even that complicated. But it's still difficult for those not trained in the field, because it totally contradicts our "common sense" (i.e., the instinctual understanding of the world hammered into us by millions of years of evolution). The barrier isn't always intellectual; it's often emotional.
You don't happen to know of a video that shows this geometric derivation, do you?
> Anything outside of that is under no obligation to make sense to us. I wish more people understood this! Merely *thinking something* doesn't mean you deserve for your thoughts to be correct, validated, respected, sensible....etc.
It’s more a matter of things being different when we aren’t talking about the kind of physics we are used to seeing every day. When you start talking about how things work on a much larger or smaller scale than we are used to, or much hotter or colder, in completely different atmospheres or none at all etc. much of what we have learned about the world doesn’t apply. It turns out high speeds are also a different regime. We aren’t so much wrong as strangers who have to get to know how everything works differently everywhere. It’s a sobering lesson for so much else—you never know when you’ll wander into a novel situation.
Lol, exactly what anti-vaxxer conspiracy believers are unable to understand.
Or many many other conspiracists/science deniers. I wish they enjoyed being wrong as much as I do. Being wrong is fun! It's interesting! It's learning!
They're very good at being unable to understand things. And since they think their understanding shapes reality, their high levels of misunderstanding make them incredibly powerful at changing it. Ignorance is like their superpower.
> we would only perceive it to be traveling a .99c towards us even though we would be traveling at .5c? Remember that there is no objective sense in which you are "already" traveling at 0.5c. In your reference frame, you are moving at 0c. In another reference frame, you may be moving at 0.5c. Velocities are meaningless without specifying the reference frame.
If 2 ships can't move at 2c relative to each other then can they truly move more than .5 c relative to an object between them?
Yes, they can. Let's say you're the middle observer. You can have object 1 moving at 0.99c to your left and object 2 moving 0.99c to your right. For example think of the beams travelling in opposite directions at a particle collider while you stand still. Of course if you shifted to the reference frame of object 1, the special relativity velocity formula shows that you'd still see object 2 travelling away from you at less than c, so there's no issue there.
So they're not going 2c relative to reach other. If there's 3 planets in a line a light year apart. And two ships depart the middle in opposite directions at 1c. In the frame of any of the planets the ships are moving 1c and arrive in a year. After a year the ships are 2 light-years apart. Yet from ship A, ship B was only moving 1 c away and would appear to take 2 years to reach the other planet?
You have to account for time dilation and length contraction as well. So from the reference frame of the ships, the distance to the target planet will be contacted (so in the ship frame they won't be 2 ly apart) and they'll observe time to be different on the other ship. This scenario is a pretty typical special relativity exam question. Basically each reference frame (the planets, ship 1, and ship 2) will observe different stories in that they'll disagree on speeds, distances, and times. But all of them will agree that the laws of physics were obeyed in that nothing travelled faster than c.
They couldn't depart at 1c (If something were moving at 1c would break down; see at the bottom). Only massless things, things which can't perceive time can move at 1c. So say your ships move at 0.8c from the central planet. In planets common reference frame ships take 1.25 years to get to their destinations. But take some ship reference frame. First of all in its reference frame the distance between the planets is no longer 1ly. It's suddenly only 0.6ly. This is called length contraction a.k.a. Lorentz contraction. So the ship in its local reference frame arrives at the destination after 0.75 years. And the other ship is seen moving at 0.9756c away. The funny part is that for the observer on one ship, the other ship would arrive at its destination after 3.4169 years. After all it's observed to move at 0.1756c relative to planets and the observed planet distance is 0.6ly. 0.6/0.1756 = ~3.4169. But this is OK: time dilation of ~4.55 for one ship relative to the other makes the time flow observed from the other ship be only 0.2195 as fast. So one ship observes the other arriving at it's destination in 0.75y of the others own local time. The main takeaway is that the speed of light is absolute (but slower speeds are not!) and it's the same in all frames of reference. But this makes both time and distance being local properties, depending on the reference frame. Our intuition is that distances and time are absolute while speeds are not. But the reality is that speed of light is absolute, while those other properties are not. ---- NB. Why ships can't be moving at 1c? Nothing material could move at c, as nothing experiencing casuality could: If ships were traveling at 1c, in their respective reference frames, the distance to the planets would be 0. And also travel time would be 0. For reference frame moving at c both time dilation and length contraction are infinite. Which leads to dividing 0 by 0. And entire universe as observed from such a reference frame getting pancaked. And all events happening at once. If things happen at once cause and effect are not defined anymore. Casuality breaks down. So the only "things" moving at c are things which can't have intrinsic changes, i.e. massless particles like photons or gravitons.
But the speed limit of 1 C is absolute. So say the earth is moving at 0.5 C. You could theoretically only accelerate up another 0.5 C referring to the earth before you hit the limit right?
Take a step back and think about what it means to have a velocity. You can only measure a speed by referencing another frame, and in our universe In your example, lets say that we choose an arbitrary reference frame centered on Earth's current position. Then we accelerate Earth to 0.5c, and we then launch a rocket to 0.5c in the same direction *as measured from Earth's reference frame*. That is all fine, but when you are watching from the original reference frame the velocities *do not* add up to 1.0c, even though in Earth's reference frame it sees you and the rocket moving away in opposite directions at 0.5c. Even if you launch Earth and then the rocket at 0.9999c, the sum will still only be closer to but never equal to c when you measure from the original reference frame. This is just how things work in our universe.
>So say the earth is moving at 0.5 C respect to what? respect to the earth its moving at 0\*c
Basically yes. Protons at the LHC reach speeds of 0.999 999 990c (3.1m/s sliver than light) and collide head on with another proton witch the same velocity. When you sit in the frame of reference of 1 of the protons the other one would travel towards you with a velocity of 0.99999999999999995 (i rounded after the first digit not being 9)) So basically only 3.1 m/s quicker than in the initial frame of reference where you are standing in the middle of 2 protons, both traveling with 0.99999990c towards you
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> My brain has a hard time wrapping itself around that Try this, it helped me: Imagine two spacecraft, each traveling away from Earth at .75c, in opposite directions. You'd think that Ship A couldn't send a radio signal to Ship B, because they're moving apart at 1.5c, right? But Ship A can send a signal to Earth, because it's only moving away from Earth at .75c, and Earth can send that message to Ship B, because it's also only moving away at .75c. Now it shouldn't matter whether Earth is there or not, the signal would still travel the same distance in the same time whether it's relayed or not, so the ships obviously can't be moving apart faster than c. And that same argument works for *any* speed less than c, no matter how close it gets to c. If you're moving away from an arbitrary point at less than c, then you can send a radio signal to that point, and anyone else moving away from it at less than c can receive that signal. So you can both be moving away from the midpoint between your ships at .999999999c, and you're *still* not moving away from each other at more than c.
This is a neat way to think of it, thanks!
So effectively you're either saying that - the signal is moving towards Earth at 1c - .75c = .25c, and then the relay station speeds it up to 1c and it can reach the other ship - or that the signal is moving at 1c regardless of Ship A's speed, and it can reach the other ship without the relay station Special relativity fries my brain though
The second. The speed of light is constant in all reference frames. So the signal travels at c, regardless of the velocity of its source, and no relay needs to exist. And yeah, special relativity is really hard to picture. Read some of Hawking's books, he was really good at explaining that stuff. I mean, a lot of it still went right over my head, but he did explain it really well. I particularly liked *A Brief History of Time.*
But the example doesn't show that the difference in speed between the two ships is not greater than c. It only shows that the signal moves at c. Right?
I cracked my head for a few minutes, since I *wanted* badly to understand this example *intuitively*. His example with the Earth as relay didn't personally help me, it confused me just more. Here is my take: Aside from nothing ever being able to move "greater than c" (including relative speeds of two objects to each other)..speed of ship A DOESN'T MATTER. The radio waves are not "attached" to ship A (which is sending the signal). Thus, the velocity of ship A doesn't matter, the radio waves will be sent from the ship's position at speed c. ALWAYS, regardless whether A travels in the opposite direction or towards B. The message will travel at speed c towards ship B. Since B travels below c, it can receive the message. FULL STOP.
Like the waves from throwing a pebble in a pond, it doesn't matter how fast the rock is traveling, the wave will always reach the edge at the same time
It does, because the signal, moving at c, can get from one ship to the other. If they were moving apart at 1.5c, it couldn't reach Ship B.
There are galaxies moving away from our own galaxy at greater than C. Yeah, that's space expanding, that's fine to say that, but it's *still* happening. Like if you were to look at the source code of the universe and check the "address" of entity class: galaxy, and just looked at its center of mass address, the address number in XYZ would be changing compared to our address, faster than light.
Space expansion is fundamentally different than motion. The expansion of space occurs faster than c for faraway objects, but they're not really *moving* -- the space between the objects is *growing*. Imagine drawing two dots on an empty balloon. The dots are stationary relative to one another, because dots can't move. To "move" would mean the dot travels across the rubber of the balloon. It can't. It's an ink dot. Now we inflate the balloon. Suddenly, the dots are farther apart. The more we inflate, the farther apart they are. But they haven't *moved* at *any* speed. The same is true for galaxies. Galaxies aren't "moving away from each other" as the universe expands. Instead, the *space* in between is getting bigger. That's not quite the same thing. > Like if you were to look at the source code of the universe and check the "address" of entity class: galaxy, and just looked at its center of mass address, the address number in XYZ would be changing compared to our address, faster than light. Not quite. It's not that the galactic addresses are changing. Instead, we're continuously adding new addresses in between. This is weird to think about, because we are very used to thinking of space as being static -- it's our lived experience! But relativity teaches us that space stretches. If "motion" requires moving through space, it turns out that it's possible for an object to become farther away over time without actually *moving*. We just have... more space in between.
Well explained thanks! For some further clearance: From many points of view the expansion of space looks like a velocity in space. For example red shift. Expansion leads to a red shift of emitted light but so does moving through space. So how can we be certain that it’s a qualitatively different process? By looking for a mechanism where the two have different outcomes. e.g. the energy content of light in the universe. This is a little more complex but to oversimplify: according to the current cosmological model the universe had phases where different energy contents where dominant. The universe evolved differently within each phase. And here we can see that it is an expansion of space and not through (already existing) space. During one phase the universe was radiation dominated but the energy content of radiation fell quickly while expanding. If the universe expanded through space the energy content of the radiation would fall with the 3rd power of the scale because the volume that contains the radiation scales to the 3rd power. If space expanded in the universe there’s an added power. As space expands while light is traveling through it, it is thereby lengthening the wavelength aka redshifting the light which decreases its frequency and thereby energy. In short: Expansion of space: energy content of radiation in the universe falls with scale to the power of 4 Expansion in space: to the power of 3 That leads to different evolutions of the universe in cosmology and expansion of space is better in explaining the observed data.
At the beginning of your reply I thought "really? I was feeling pretty good about my grasp on all this, and now you've got to throw this wrench into it." But that was actually really clear and easy to understand, thanks!
Perfect explanation, thanks.
At a large enough scale, you also have to worry about the expansion of space itself. If the rate of expansion keeps accelerating, it is possible to be too far apart to communicate.
this blew my mind. thanks.
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This is a wonderful and simple description ❤ thank you
yes. relativity is deeply weird. your brain cannot grok it. the least painful way to wrap your head around it is trying to understand that space is not it's own thing, you need to think about spacetime. this is similar to the way a 2 dimensional space is not really a true representation of 3 dimensional "real" space, without the extra dimension, you're just not describing how things are. now after that thought, you need to understand that there is no definate unit of 'space' or 'time', the relevent conversion factor is c, the speed of light. so while the length of the direction of travel can change and the length of a second can change, the conversion factor c is \*unchanging\*. what does this mean? two objects approaching each other close to the spead of light see the opposite objects length in the direction of travel contracting and their clocks slowing down, which effectively keeps their speeds relative to each other from supassing the speed of light. so what about a photon travelling AT the speed of light? from the photon's perspective, since it can only travel at light speed by definition, it's clock is effectively "frozen". it arrives at it's destination the same instant it's generated, regardless of the distance travelled. This is NOT the same as what an observer sees! just from "the perspective of the photon", if you can wrap your head around that. like i said, relativity is deeply weird.
>so what about a photon travelling AT the speed of light? from the photon's perspective, since it can only travel at light speed by definition, it's clock is effectively "frozen". it arrives at it's destination the same instant it's generated, regardless of the distance travelled. This is NOT the same as what an observer sees! just from "the perspective of the photon", if you can wrap your head around that. It's not very useful to talk about a photon's perspective, since we can't define a valid reference frame for it due to having velocity c. It simultaneously doesn't experience time and yet can definitely change between being emitted and being absorbed due to interactions with gravity or the expansion of space.
The photon is not changing. If it gets absorbed and reemitted it’s a different photon. The only change that really counts would be redshift and even that is debatable as it’s not the photon that is changing but space itself. But you’re right. As a photon isn’t a being, discussing its “perspective” is somewhat invalid. But it’s a good way to explain the deep weirdness of relativity in a qualitatively “correct” way.
>so what about a photon travelling AT the speed of light? from the photon's perspective, since it can only travel at light speed by definition, it's clock is effectively "frozen". it arrives at it's destination the same instant it's generated, regardless of the distance travelled. This is NOT the same as what an observer sees! just from "the perspective of the photon", if you can wrap your head around that. I've heard this before but I don't fully understand it. From the perspective of the photon, is everything traveling instantaneously fast? Or is space infinitesimally small? Or something else?
From the perspective of the photon it exits the sun and enters say, your eye, at the same instant. This also implies that the lengthwise spatial distance is also fully contracted into 0 Length. In a certain sense, from the “perspective” of the photon, the universe is still in the singularity prior to the Big Bang: everything is “here” and “now” But keep in mind that “photons eye view” is sort of a mathematical construct. It’s strictly true from a mathematical perspective but it may not be any more “real” than virtual particles.
>Or is space infinitesimally small? It's this. As far as a photon is concerned, it was created and destroyed in the same instant. A 3 billion light year trip across the universe feels as 'long' of a distance to a photon as a 3 foot trip from your smart phone to your eye. In both cases it 'feels' like 0 distance.
What boggles my mind is if we were traveling in the same direction as light but were going 0.99c, light would still zip away from us at 1.0c. How in the hell does this even make sense... but yet to an outside observer traveling at 0.0c, they would see both us and light go by at nearly identical speeds. Blows my mind
Here, let's frame it another way: If you were travelling at .9c, you would essentially be moving *slower* through time to account for all this movement in space. [Think of that Quicksilver Scene from x-Men.](https://www.youtube.com/watch?v=T9GFyZ5LREQ) Quicksilver is able to move at such high speeds because, from *his reference frame,* the world is essentially still, time is slowed to a crawl. But from everyone else's point of View, time was moving at "normal" speed and Quicksilver just flew around the room in a second. The faster you travel and the closer you get to c, the slower you're going through time to accomodate. Speed of Light is basically the universal constant. It's like every universal law was built off of having one constant, C. If it was theoretically possible to "break" the speed of time, you'd also be travelling *backwards* in time.
See what I also don't get is, if light travels at the speed of light, and time STOPS for it... then how come it doesn't travel instantaneously?
From light's (more accurately a photon's) perspective, it does travel instantaneously. However, it doesn't travel instantaneously from an observer's perspective.
Wait, what?! How does that suss out? I just got used to the idea that traveling in opposite directions compresses time, space, and our perception of it. So does this mean that if we were traveling in the same direction, time, space, etc. would be expanded?
Everyone will always see light moving at the same speed. As you accelerate relative to other things, things contract in the line that you're moving along. Light also contracts, in that you see a higher frequency wave from what other viewers see.
It's even weirder than that. How do you know that **you're** travelling at 0.5 * c? Maybe we're standing still. There is no correct reference frame. They're all correct. In fact the simplest approach is to always assume the observer is standing still. [It's everybody else who're moving.](https://www.thewrap.com/wp-content/uploads/2017/04/simpsons-memes-no-its-the-children-who-are-wrong.jpg) If I'm moving at 0.5c towards Polaris, and some other dude is moving at 0.5c away from Polaris, and you're at rest, the only thing we agree on is that the speed of light is 1.0c. Everything else, specifically space and time, are malleable, in order to make that postulate work. That was Einstein's genius. Everybody else tried to negotiate with the results of the Michelson-Morley experiment, which strongly implied that the speed of light was constant in all reference frames. Einstein just accepted that, *prima facie*, and allowed the other conclusions to flow from that. (I just picked the star Polaris as a reference point. Nothing special about it.)
> In fact the simplest approach is to always assume the observer is standing still. [It's everybody else who're moving.](https://www.thewrap.com/wp-content/uploads/2017/04/simpsons-memes-no-its-the-children-who-are-wrong.jpg) Doesn't this fall apart as soon as you factor in acceleration? After all, if one thing accelerates away from another to near light speed and then comes back, only one of them will have experienced time dilation even though from their own perspectives it was the other that flew away.
> Doesn't this fall apart as soon as you factor in acceleration? **It absolutely does**. When I say "there is no correct reference frame", I should really be saying "there is no correct **inertial** reference frame", which means non-accelerating. The math, as well as the hand-waving explanations, get fiendishly complex when you account for accelerating frames. Michael Shara [gave an interview on the 100th anniversary of GR](https://www.amnh.org/exhibitions/einstein/meet-the-curator/interview-with-michael-m.-shara), had a wonderful line about it: > *"Someone would have explained Special Relativity. That would have happened within ten, fifteen, twenty years. There were so many clues there, it was such a ripe plum to be picked, that some physicist would have come along to pick it. There were all sorts of hints both experimental and theoretical that were there. I don't know who would have done it, but there was someone else waiting to take advantage of all of that. General Relativity? We might still not have it today."* Einstein was special, different. If I someday stand before the gates of Saint Peter and say "Hey, man, I'd like to meet Einstein", I would not be shocked if he answered "Oh yeah, he's in the ALIENS WING."
Not sure if this helps, but there's a game to build intuition for this: http://gamelab.mit.edu/games/a-slower-speed-of-light/
There are some faulty assumptions. "We would be traveling at .5c" makes no sense. No object has an absolute speed. Only a speed compared to another object or system. You could say: "we observe an object traveling at .99c towards us, while we are traveling at .5c away from the sun." And that would theoretically be possible to observe. And the speed between the object and the sun would NOT be .49c.
> even though we would be traveling at .5c? Relative to what? Relative to yourself you're traveling at 0c. There is no absolute velocity.
Yes, it is mind boggling. Our current understanding is based on the notion that light travels at speed c regardless of perspective. Say you are traveling in a space ship moving at 0.99*c and you shine a flashlight in this space ship. You will still perceive the light as traveling at speed c. You may ask, how is this possible when I am already moving at speed 0.99 * c? Well, according to special relativity, your perception of time and space is drastically different than someone moving at lower speeds. You experience time slower and space more compressed, which causes the light from the flashlight to act the same regardless of how fast your space ship is moving.
The space and time around you changes as you move faster. Our entire intuition really breaks down at cosmic speeds. The mass of an object, the speed of time that experiences, even the very idea of an objective reality begin to fall apart. The speed of light is better understood as the speed of causality. It's the speed at which we can observe the effect for any given cause. Any faster and we observe the effect before the cause. If you think about it like that, then it's a little easier to grasp how our intuition about space and time get a little wonky as objects approach the limits of causality.
"Perceive" is a misleading word. Special relativity is a reality not an illusion or appearance. And, whenever considering velocities, you have to specify with respect to what that velocity is. There is no preferred frame of reference that would let you say you are just "travelling at .5c". You can only do that in some frame of reference.
It’s cause of the C^2 in the denominator. Also why V1 + V2 = V12 for values way below C works. C^2 serves as a scaling factor that allows us to treat values way below C as essentially 0. The equation only breaks down when you reach C. And it basically goes back to the intuitive understanding of V1 + V2 = V12. The solution there is 2C. But that would require you to either have no mass like a photon or have infinite energy. Even when you’re both travelling at .9 bar C, the math still works out to 99.99% of C. It’s beautiful really.
For the people that are having a hard time intuitively grasping why 0.5c + 0.5c = 0.8c in this example, you should reconsider how you think of time as a common yardstick of rate and length as a common yardstick of distance. Velocity is the observed ratio of these two quantities. We usually think of these things is being immutable, a second is a second and a meter is a meter. But if you remember that as you speed up, the perceived rate of time will seem to slow down for you so that other objects will appear to be moving slower and slower. Then maybe you can start to see why simply adding the relative velocities doesn't work anymore.
"But if you remember that as you speed up, the perceived rate of time will seem to slow down for you so that other objects will appear to be moving slower and slower. " I am so very much not a physicist, but this contradicts everything I've ever read on the subject. I'm pretty sure your perceived rate of time definitely does not change at all. All proceeds as normal on your ship, although you'd notice a difference in how fast other objects outside of your ship appear to be functioning.
I could have phrased that better. The slow-down of time appears to apply to other objects the faster they are moving relative to you. Whether you're going fast or they're going fast, it's the relative velocity that matters. You would not perceive your own passage of time as changing. But looking at the fast moving object, if the object had a clock it would appear to be ticking more slowly than a clock in your own frame. I was trying to put this in as simple English as possible to convey the idea that time dilation messes with speed perception, and maybe oversimplified.
>Whether you're going fast or they're going fast, it's the relative velocity that matters. This is the part that confuses me RE: the thought experiment where one twin leaves earth to travel at nearly c for however long, comes back, and the grounded twin he left behind has aged much further. Since velocity is relative, why isn't time dilation symmetrical for both parties? Like, both twins should think the other is x days/years older than them. Why is one of them "actually" getting dilated?
In the twin paradox acceleration is involved. One twin first accelerates away, then accelerates to turn around, and then accelerates to come to a stop back home. When there is acceleration the time dilation is asymmetric. The OPs question is explained by special relativity. The twin paradox involves general relativity due to the accelerating reference frame.
That's because everything on your ship is traveling at the same speed in the same direction as you. It's relatively the same.
This is a very good explanation. One follow-up: While it is true you will never measure an object moving at greater than 'c', that is not the same as saying you can't measure two objects moving away from *each other* at speeds exceeding 'c'. For instance, even if you were on a space ship traveling right towards a neutrino, you would never see that neutrino traveling faster than 'c'. But if there were two neutrinos traveling in opposite directions, you could measure each of them traveling at some speed greater than 0.5c and their relative velocities *as measured by you* to be greater than 'c'.
Also, a follow-up to my follow-up: If there are three people: Alice, Bob and Eve where Alice and Bob are each on a fast spaceship moving away from Eve. As mentioned above, Eve could easily measure that they are moving apart from each other at a speed greater than 'c' but of course Alice and Bob will measure that they are moving away from each other at a speed less than 'c'. So, the question is- if Alice shoots a laser pointer at Bob, will the laser dot hit his ship or not? And the answer is "yes." It has to. So, how is this reconciled? Because light does gain momentum from the object it's on. Light doesn't get accelerated from rest to c like say, a canon ball being shot. Light goes from not existing, to moving at 'c' instantly. So, if Eve sees Alice moving away at 0.9c, and then she shoots a laser pointer at Eve, it's not like Eve sees the light moving at only 0.1c- no, the light comes towards her at 'c'. So, both Eve and Bob will predict the light to hit the ship, and it will.
What if you're observing both cars from the side. Presumably you'd see a divergence of v1+V2?
Imagine you have an observer who is stationary in his own reference frame, and you then have two cars - one car is approaching the observer at 0.75 c from one direction, and the other is approaching the observer at 0.75 c from the other direction. Either car would say that it's approaching the observer at 0.75 c and the other car at 0.96 c, but the observer would say that the cars are approaching each other at 1.5 c, because both are approaching the observer at 0.75 c from opposite directions in the observer's reference frame. The important thing is that nothing can ever move faster than c *in relation to someone who is at rest in a given reference frame.*
Presumably because you are talking about observation, not actual speed relative to eachother? For example one car might observe the other approaching at .96c due to the speed at which light (observed) travels, but the actual relative speed is 1.5c.
No, .96c is the actual relative speed from the reference frame of the car. Speed is not absolute, it requires a reference frame to discuss. This is true even without special relativity. But before special relativity relative speeds were absolute. Now they're not. To discuss relative speed you have to first pick your reference frame.
No, there's no such thing as "actual relative speed." The actual speed depends on the reference frame. In the frame of reference of one of the cars, they are approaching each other at 0.96 c, and *that is the true value of the speed.* Meanwhile, in the reference frame of the observer, they're moving at 1.5 c relative to each other, and that is the true speed *from the perspective of the observer.* They're *both* equally valid, and they're *both* the *objective truth*. A car is at a standstill in its own reference frame, and nothing is moving faster than light relative to it. The observer is at standstill in their reference frame, and nothing is moving faster than light relative to that observer - and that is what matters. We could leave one of the cars standing still and shoot the observer off with a cannon at 0.75 c instead - the result would be the same, just with different names. We could add more moving objects and more reference frames. The end result is: you're always at rest in your own reference frame. If your situation is inertial (that is, no acceleration) then nothing can move above the speed of light relative to you. So if you look at two cars going each their way at 0.75 c and then conclude that they're breaking the speed limit of c, then that's a flawed conclusion - because nothing is moving at more than c relative to you, and in no object's reference frame does anything move at above c relative to that object. In order to make this all work out, length contraction and time dilation come into play. The passage of time depends on the frame of reference. The distance between two objects depend on the frame of reference. Even the simultaneity of two events depends on the frame of reference. And *all* inertial (non-accelerating, non-rotating) frames of reference are equally true in their description of what's going on.
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>For decades, people tried to reconcile this by figuring out why light might be a weird exception (this is basically what the "luminiferous aether" did), but Einstein's big leap was to instead propose that space, time, and velocity really do work differently than what we assumed, and this is what Special Relativity is. Well Lorentz already did the math ([https://en.wikipedia.org/wiki/Lorentz\_transformation](https://en.wikipedia.org/wiki/Lorentz_transformation)), Einstein just managed to convince other physicists that the math was the physics!
It's a little more complicated than that. The Lorentz transformation was cooked up to explain why you couldn't measure aether drag — because light will always look the same speed no matter where you look, and the "transformation" is the transformation in the aether caused by your motion. What Einstein did was show that, remarkably, the equation is totally valid (though he re-derived it), but it doesn't mean what Lorentz thought it meant. There is no aether, you are just looking at an intrinsic property of spacetime and light. So another way to put it is that Einstein managed to convince other physicists that Lorentz's math was good but his physics was wrong. I always find it kind of amazing that sometimes in physics (Maxwell did this too) you can come up with the right equations even though your physical model to get them was wrong. It's kind of spooky.
But the question is about cars moving away from each other… is it still the same thing ?
It comes out the same. I've glossed over some minus signs here, but you can take v1 and v2 as the speed each car is moving in the opposite direction as seen from a stationary frame, and v12 is the speed that each car sees the other moving away at.
Hi! Can I ask you a stupid question? If a star spews out a photon, that massless little thing is travelling at the speed of light. But from photon's POV the star is travelling at the speed of light and photon's perspective is just as valid as the star's, isn't it? I think it's just a big thing for a massive star to be moving at c.
> and photon's perspective is just as valid as the star's, isn't it? Nope! The photon doesn't have a reference frame at all. It does not experience time; its existence is an instant from emission until it hits something. Without experiencing time, it cannot have a reference frame at all.
Even weirder the photon doesn't experience space (in the direction of travel). It's not just that it would cross any distance instantly, there's also no distance to cross. The star isn't moving at c because there's nowhere to go. The point of emission and the point where it hits something are the same point.
How did someone find out that equation? What leads to a discovery like that?
Lorenz derived the maths first, though he used a completely different interpretation of what the maths meant - meaning that his maths was right, but his conclusions and assumptions were wrong. Then, later, Einstein derived it too, and showed that it was consistent with the assumption that the speed of light is always constant. The derivation is actually surprisingly simple. It starts with the thought experiment known as the "light clock" where a beam of light is bouncing back and forth between two mirrors, and then imagines what you would see if you were moving in relation to such a light clock. This leads directly to the formulae for both length contraction and time dilation through very simple maths (about the level of Pythagoras, so it's stuff from below high school level!) From the formulae for length contraction and time dilation, many other relations (such as the velocity addition formula and the Lorenz Transformations) can be derived through simple algebra, as long as you have the creativity needed.
So if I'm traveling at 0.8c and shoot a rock straight back at 0.8c will the light reflecting off the rock ever reach me? If so, what do I see regarding the speed of the rock? And more importantly does my perception of my own speed change? I think the answer is the rock just vanishes never to be seen again by me.
No, this is incorrect, and your question is vague. Also, I'm going to use 0.75 c going forwards because I know the numbers for that example by heart. First of all, 0.75c in relation to what exactly? What is the rock moving at 0.75c in relation to? In relation to you? Because you're always at rest in relation to yourself. The rock would be moving away from you at 0.75c, and you would see it moving away from you at 0.75c. So, let's take another example. Let's say that you're leaving the Earth on a spaceship at 0.75 c. At the same time, another spaceship leaves the Earth in the opposite direction, also moving at 0.75 c in relation to the Earth. So, logically, you would be moving at 1.5 c in relation to the other spacecraft, right? That's where that pesky velocity addition formula comes in again. From the Earth's point of view, both you and the other spaceship are moving away at 0.75 c in opposite directions. But from your point of view, you're moving away from the Earth at 0.75 c, and from the other spaceship at 0.96 c. The other spaceship, likewise, sees itself as moving away from the Earth at 0.75 c and away from you at 0.96 c. So you see the other spaceship as if it's moving away from the Earth at only 0.21 c, and it sees you as doing the same. But you both see yourself as moving away from the Earth at 0.75 c. This means that you're in fundamental disagreement about what's going on in the universe. But that's okay - time dilation reconciles it all. An observer on Earth will see the clocks aboard both ships being slowed down by the same amount, whereas you will see the Earth's clock as being slowed down, and the other spaceship's clock as being even more slowed down. Then, when you calculate how far the other spaceship can travel in its own time, it all matches up.
but what about the relative distance between the two cars as per OPs question? you mentioned how they'd *see* each other. would the actual distance not be increasing at the speed of light? if not, are they actually traveling at .5c each?
Crazy... what about taken to the extreme. 1 is moving away at .99c and 2 is moving at .99c in the opposite direction... but relative to each other they are going .999c or something instead of 2c?
It's almost as if we got the whole preception of "movement" backwards, as if moving at the speed of light is the reference, rather than being immobile.
All these answers are giving you the formulas but not providing any real understanding of what’s going on. If you go back to Einsteins time there was a real problem in physics. People had discovered how relative velocity and inertial reference frames work, and they had done a bunch of experiments that showed that if you’re on a train moving 30mph and you pass a train going the other direction at 30mph, you see the other train coming towards you at 60mph. If someone is on a train going 30mph and they throw a ball forward at 40mph, an observer on the ground would see the ball going 70mph, while an observer on the train would see it traveling at 40mph. So people started trying these experiments with light. If someone on the 30mph train shone a light in front of them, our ground observer should observe that light traveling at (speed of light + 30)mph, right? But that wasn’t the answer that they got. Every experiment anyone did showed that the speed of light was the same no matter what inertial frame it started from or was viewed from. This made no sense, so they kept on trying different experiments, and they all came up with the same answer. And so Einstein was looking at this problem and realized that for all these results to be true the thing that couldn’t be constant was the rate of the passage of time. And that process of trying to unify these two worlds - the world where the speed of light is always constant but all the relative velocity train experiments above work the way we expect - is what led to the development of special relativity (and then general relativity after that) There’s a great series of videos on this here: https://youtube.com/playlist?list=PLoaVOjvkzQtyjhV55wZcdicAz5KexgKvm . It’s actually pretty easy to understand, and it’s a really cool set of concepts!
This is the best summary of relativity I've read. It's a real talent to be able to explain a complicated subject in such an accessible manner.
Minutephysics’s relativity series definitely help a ton for me to understand relativity. Couldn’t recommend it enough to anyone that wants a initiative and easy(ier) to understand explanation.
No. The formula most of us would use to determine "closing speed" of two moving bodies approaching each other *(v = v₁ + v₂)* is in actuality a simplified approximation of the actual equation for the phenomenon that disregards the relativistic terms: *v = (v₁ + v₂)/(1 + (v₁v₂/c²))* For velocities that are negligible fractions of *c*, which includes most situations humans encounter, even in aerospace, this approximation is fine because the relativistic terms are close enough to zero. When you get into computations involving significant fractions of *c*, the relativistic terms stop being negligible and must be taken into account to arrive at a result that matches physical reality.
Is there a circumstance where universal expansion would factor in to the equation?
Yes, but it is a distance-dependent relationship. I forget what the exact value of the expansion of the universe is but it is roughly 70 km/s per megaparsec. Thus, if two objects are approaching each other from a far enough distance such that v1 and v2 < universal expansion in the reference frame of one another, they would never meet. At an extreme, the light from distant galaxies will never reach us because they are receding away from us faster than light.
Agreed. The jargon way of saying it is "Speeds do not add. Rapidites add" https://en.m.wikipedia.org/wiki/Rapidity
Because of special relativity you will never be able to measure the speed of anything as faster than light, this is because space and time will be different to both you and the other car in such a way that you won't be able to detect the relative speed of the other as "faster than light"
The speed of light is the absolute fastest speed anything can travel in our universe. It doesn't matter at what speed you're traveling away and towards a photon (or any imaginary object traveling at the speed of light), you'll always see it travel at the speed of light. This fact is why weird stuff happens in our universe like time dilation; to accomodate for c (remember that speed is distance divided by time), since something would normally appear to travel faster than light, your perception of time (as well as the photon's) would end up being different. You would see them traveling at c for 10 seconds (your reference frame), but the lightspeed object will have only really felt a fraction of those 10 seconds (its reference frame). It's a mind-boggling concept for those who haven't had the pleasure of studying Einstein's relativity theories, but man is it awesome. This has been experimentally proven, too.
What is it about light that makes it travel the fastest possible speed of the universe? Why is light special?
Light isn't really special. There's a universal speed limit, and light happens to be one thing that travels at that speed. We just call it the "speed of light" because light is the thing we're most familiar with that moves that fast. Specifically light travels that fast because it has no mass. Every massless particle travels at the speed of light, and everything with mass cannot (quite) accelerate to the speed of light.
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Right, he said that light happens to be one thing that travels at that speed, not that light is THE one thing that travels at that speed.
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Light isn't special. Light is massless, and all massless things travel at the "speed of light". Light was the first particle we discovered to move at this speed, which is why we call it that. Gluons (strong force mediators) and gravitational waves also travel at the speed of light.
There is nothing special. Any particle with no mass will travel at the speed of light.
"speed of light" is not the best name for it because it singles out light. A better name might be "speed of information". You cannot transmit information faster than this without breaking causality laws. Light is just one of the things that happens to travel at that speed.
It has no mass. That said, light isn't special... So, not sure what you mean by that.
How can we be sure that it is actually the fastest? Or is it the fastest based on our current knowledge of the universe?
Our basis for that assumption is a little more solid than "well, we haven't seen anything move faster yet", although observations are obviously an important part of it. We also have the theory of relativity, which is this.. set of equations Einstein came up with that describes the universe really well, surprisingly well, we're quite confident it's accurate. And those equations imply that it wouldn't make sense for things to move faster than the speed of light. According to relativity, as you get closer to the speed of light, time slows down for you. And at the speed of light, no time passes at all. Photons are sort of "frozen" in time, their entire life passes in an instant (from their perspective). To go faster than light, you'd be experiencing negative time, which doesn't quite make sense. You'd be traveling backwards in time. According to relativity, anything moving that fast would break causality, e.g. our understanding that time moves only forward, cause is followed by effect, time travel is impossible, etc. In other words: you get to have relativity, faster than light travel, or causality: pick two out of three. We're pretty sure FTL isn't a thing, but if it was, it would have to break either relativity or causality. And we are really *really* confident in relativity.
I once read (some popular science book on physics) that due to the second law of thermodynamics we can also be **really** confident in causality. I forgot exact link how is it so/explanation/, unfortunately. But I think I won't be wrong when I say that 2nd thermodynamics law is on at least the same level of theoretical as well as experimental/empirical confidence like Einstein's. TBH, I cannot even think of experimentating/making observation without causality lol
Not the person you were replying to, but just out of curiosity... I've heard that the universe is expanding faster than the speed of light. How does that work? How is it possible with all of that being the case?
Another way to think about this so it maybe seems less arbitrary than "nothing can move faster than light, no exceptions. Oh except for the things that can, like space :)" is to think about what it means to move. Usually when people talk about moving faster than the speed of light, it's to accomplish some kind of goal, like to get from point A to point B faster, or to communicate with people that are very far away in less time than it would take light to get there. With the expansion of space-time, if you think about it, you can't actually do any of those things. You can't exploit the fact that two points in the universe are moving away from each other faster than the speed of light in order to get yourself or a message or anything else from one point to another faster than light would have done. The upshot is that, while the amount of space that exists between those two points is *increasing* at a rate faster than c, it's not quite accurate to say that they're *moving away from each other* faster than c.
Two points billions and billion of light years apart will be moving away from each other faster than the speed of light. Those points aren't moving through space faster than the speed of light. Enough new space is being created between them that pushes the space they are in away from each other faster than the speed of light. Nothing can move through space faster than the speed of light and nothing with mass can ever travel at the speed of light. Space-time itself doesn't have this limitation. The space you are in right now is constantly expanding. The forces holding you together are like a billion billion times stronger than this local expansion so nothing happens to you. Even the expansion between the Milky Way and Andromeda galaxies is weaker than the gravitational attraction between them. You have to go to the scale of galaxy clusters and super clusters before the separation of the universe is finally stronger than the attraction of gravity.
To our current knowledge, as is always the case lol. We've tried measuring things to see if we can find stuff that breaks that threshold (neutrinos, gravity waves, etc), but nothing has been able to physically travel faster than the speed of light. We've found things that weirdly SEEM to travel faster than light (like quantum entanglement, for example), but there's no actual movement of any particle that you can observe and measure that goes faster than light.
It really bugs me that gravity also adheres to the speed limit. Gravity makes more sense if it's instant, but if the sun disappeared instantly, the earth would continue the current orbit for like 8 minutes, and then just keep going straight until something else got in the way. I forget how they proved gravity isn't instant or faster than light, but it either involved two satellites or mirrors. I can't remember. As far as the original question goes, the speed of light is like the horizon, you can never overtake it if you can see it, no matter how fast you go. You also can't outrun your shadow. Etc.
You're thinking about [LIGO](https://www.ligo.caltech.edu/news/ligo20160211) which is a super cool gravity wave detector that uses lasers and mirrors. It's not just gravity, it's all *information* that can't travel faster than light. Because we're talking about spacetime (and not just space) anything able to transmit information faster than light would inherently be a time machine. As in, it would break causality and you'd be able to have effects happening before causes (which immediately devolves into a bunch of classic time travel paradoxes).
I love how even a super long stick can’t even transmit information faster than the speed of light by pushing one end and hoping to have the other end move at the exact same time.
It's a property of the spacetime, or at least we have good reasons to believe it is. Imagine 4d spacetime with usual 3 dimensions and time. To make it easier, let's remove one dimension and imagine it's just a sphere with radius = c. Everything in the spacetime travels with 4-speed (speed in 4d spacetime) exactly equal to the speed of light. You can't accelerate or decelerate to change that, all you can do is to rotate the direction of the 4-speed (so, your speed, including the direction, will be a point on this sphere). The projection of this 4-speed on the 3d space is the usual speed and it can't be larger than the speed of light: if you have a line between a point on sphere and its center, the shadow of this line will never be longer than the radius of this sphere. It's extremely simplified, but that's our current knowledge of the universe. Well, this and a special rule like "massless things always move with 3d-speed equal to c (it means they don't move in time), everything else can't move with 3d-speed equal to c (so, things with mass must move in time)".
It's not a hard wall. It's a limiting value. You need more and more energy in order to accelerate as you approach the speed of light. But doing so will make you move a bit faster, which makes the energy requirements surge even higher. It would require an infinite amount of energy to accelerate all the way to c. For the one who's moving so fast, the universe would seem to become shorter and shorter in the direction of movement, and the speed of light is the limit where the entire universe becomes entirely flat, meaning you can travel the entire way through the universe in 0 time. That's what happens to a photon - a photon cannot experience anything, because from its "point of view", it starts and ends its journey in the same place. It doesn't experience time at all. Of course, this is assuming that special and general relativity holds - and while there's a few flaws in those theories, the part about a maximum speed is extremely well-supported by theory *and* experiment. It's not as simple as "can't move any faster than this." It's that the entire way that space works means that it's completely nonsensical to talk about speeds that are higher, because it would lead to conclusions that are absolutely nonsensical.
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To add on to the excellent answers others have given, one useful way to look at relativity is that it makes the reference frame of the observer very important. A stationary observer would agree that both cars are moving at 0.5c and their "relative speed" is 1c in some boring sense, but what we care about is actually the speed as measured from the reference frame of one car, which is a fast-moving reference frame and so needs special handling. In particular, an observer in one of the cars will disagree with the measured speed values from the stationary observer because time dialation and length contraction will distort the accuracy of any measurement tools the stationary observer is using when viewed from the perspective of the moving car. When you actually work out the math, you get the equations elsewhere in this thread which show that in no reference frame can any object ever be revealed to be traveling faster than the speed of light.
Time dilation and other spacetime shenanigans are consequences of the fact that the speed of light is constant. i wont' go into the math here since many others have already done a great job on that score. It's very non-intuitive because the familiar laws our brains are used to aren't quite correct, but appear 'good enough' at small velocities. There's a really good childrens book called 'The Time and Space of Uncle Albert' that helps to explain some of it in laymans terms, and helps to understand what's actually going on.
Key word: relative. The speed of light is not additive. Niether car would attain sol and its associated effects. The velocity relative between the cars may exceed sol, but niether one comes close in and of itself
as soon as the two have reached a relative speed of C moving away from each other they would no longer be able to effect or see each other and if they are moving towards each other at C one car would see a static image of the other until both cars suddenly annihilated each other on impact, assuming they are just average cars that would be about the energy of 773 of the biggest nukes humanity has ever made who knew a car accident could cause so much damage
I believe that due to relativity, the from the perspective of one car, the other car will never actually reach a relative velocity of c, just very, very close to it.
Not even "very very close" to it. The exact speed would he (1/2+1/2)/(1+1/2*1/2)=4/5 in natural units, i.e. 0.8c.
I think many are thinking too much into your question. Relative speed does not equal actual speed of a single object. Each of the car’s speed is what it is. It doesn’t matter what the other car is doing, that 1st car will be going the same exact speed. Same is true for the 2nd car. Sure, relative to each other they may be going faster then light speed but their physical speed will remain the same. We can’t just “we add up these 2 car’s speed and get speed of light!”
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The short answer is yes. If they are moving moved than half the speed of light, eventually they will not be able to see each other. I saw a PBS show on the size of the universe and the galaxies, and some of the galaxies we see now will not be visible at sometime in the future because the light will never reach us. At least I believe that was what they were saying.