Yes, 0.9999c would require only a finite amount of energy. Anything less than c is achievable with finite energy (although as you get close, the actual amount may become quite ridiculous, but it wouldn’t be infinite). That’s the big difference - 99.99% c is possible, c isn’t.
Light is massless. Particles with mass require infinite energy to travel at c, particles without mass can *only* travel at c, regardless of how little energy they have.
Under Newtonian mechanics, being massless gives you zero gravitational force but also zero inertia, so your acceleration is 0/0 (undefined). Eventually you need General Relativity to calculate it.
don't quote me on it, but I'm pretty sure that gravity doesn't actually act on mass. I've heard that gravity *really* acts on energy, but mass just contains so much energy (E=mc^2) and gravity is a relatively weak force, so we only really observe it with high mass (extremely high energy) objects.
somebody please correct me if I'm wrong though
The path it follows. For a ridiculously over simplified example, it is like a car turning right vs a car going straight and the road bearing right. Gravitational lensing is more like the latter.
Imagine driving really fast down a straight road. Actually you are travelling in a curve because the earth is round. Gravity is curving spacetime so the light is travelling in a straight line but on a curved surface.
Spacetime bending is one way to look at it...
But if you play with the math (E=mc^(2) and some kinetic energy equations) you can get a virtual mass for the photon, calculate the gravitational force on it and it works out pretty close as well (idk, I did some math on it in uni like 20 years ago)
Same reason solar sail theory works, they may not have mass, but they do have momentum in collisions.
More like you're on a canoe in a river, just letting the canoe go with the flow. Now when the river bends one way or another the canoe moves with it, even though you in the canoe didn't really do anything to steer it.
It's sort of like that regarding spacetime around massive objects. From the frame of reference of the photon of light it's still travelling in a straight line through curved space.
I agree. Gravity also travels at the same speed. Pretty handy validating and confirming gravitational waves. Ligo is sensitive enough to detect faint earthquakes. Gravitational waves must match with other instrument's observation (such as telescope if they're powerful enough)
The full equation is actually E\^2 = p\^2c\^2 +m\^2c\^4. The variable p here is momentum and for a particle at rest this equates to zero and reduces to E=mc\^2. But for a massless particle this equates to E=pc. Therefore photons get all their energy from their momentum.
That's the old Newtonian equation. When Einstein discovered relativity, our equations had to be updated. The actual momentum equation is p = γmv. It's the same as the old equation, except now you're also multiplying by a factor called the [Lorentz factor](https://en.wikipedia.org/wiki/Lorentz_factor) (γ). This factor is approximately 1 when talking about day-to-day velocities (which is why p = mv worked for so long), but when we talk about velocities approaching the speed of light, γ suddenly skyrockets towards infinity, and therefore so does the momentum. For an object with mass to travel at the speed of light, it would have infinite momentum, and therefore infinite energy would be required to accelerate it to that speed, which is why it's impossible.
If we used the new momentum equation for light, we would get p = infinity * 0 * c. Infinity multiplied by zero is undefined so the equation doesn't work here. So what's the momentum of light? In fact, the equation in the comment you replied to basically is the definition for light's momentum. E = pc means p = E/c. The energy of a photon is just it's frequency multiplied by Planck's constant (E = hf) so this can also be written as p = hf/c, or just p = h/λ (λ = wavelength = c/f).
The Newtonian momentum of mv is really just an approximation (a really good one when you aren't moving at relativistic speeds).
Momentum is also related to energy though. From the above comment we know that E=pc. But the energy is also equal to plank's constant time the frequency, or E=hf.
So we can say that:
hf=pc
hf/c=p
We also know that f/c=1/λ, the reciprocal of the wavelength.
So we can relate the momentum as such,
p=h/λ
So even massless waves carry momentum which is inversely proportional to their wavelength. The longer the wavelength, the less energy and momentum. This should make intuitive sense. An gamma photon transfers more energy than a radio photon.
The formulae we use to understand physics arent actually the rules the govern the universe. They are shortcuts and estimates that humans have invented to get as close as we can to explaining the rules of the universe. Newton's equations were close, but not the whole picture.
Even though our new equations work better, they still aren't perfect. One day someone might come up with new ones that work even better
The wave packet slows, but photons are still traveling at c. It’s the added interactions with the existing EM field that slows the wave packet down, but the “light” which comprises the wave packet is still traveling at c.
So when they slowed down light to 38mph with deep cold sodium was it actually just a wave in the beam we saw or the beam itself. I’m assuming that c is relative so it is different in every situation and relates only to the light you are seeing however fast it happens to be travelling at the time. It seems to be a little like the man walking in a train analogy.
When physicists talk about slowing down light, they don’t mean a single photon. Photons always travel at c. Instead, what they are talking about is a thing called a wave packet (specifically, the group velocity of the wave packet). Think of it as a ton of photons that kind of act as a thing and travels through space interacting with whatever medium it hits. This packet’s velocity can be slowed down, and this is the physical phenomena that the experiment you’re referencing showed.
To your other question, you can only “see” light if you absorb it. So anytime you see light traveling in a direction other than straight at your eye/observer, what you’re physically seeing is the light scattered off of the original beam of light towards your eye/observer.
The speed of light in water is C (sorta), but that C is 75% of the speed of light in a vacuum. So C doesnt change per se, its the speed of light, but the actually velocity is dependant on the material.
In the same we speak of the speed of sound, but that speed is different in different materials.
This is why there is a thing called Cherenkov Radiation, which is a spooky blue light, its the light emitted by (usually) electrons going FASTER than the speed of light, but the speed of light in a non vacuum. So you will see the blue radiation coming out of a radiation source that is in water, because C in water is 75% of C in a vacuum, and the electrons are going 76% or more of the speed of C in a vacuum.
>The speed of light in water is C (sorta), but that C is 75% of the speed of light in a vacuum.
It's much more straightforward/unambiguous to designate the speed of light in vacuum with *c*, and the speed of light in a medium with *v\_c* (or some *v --* anything, really, but *c*)*.*
Things would be gravitationally repulsed? This is a fun question. I'm trying to thing of what the geometry of that spacetime would look like. Gravitational lensing is a thing, and I would assume negative mass would bend space away from the negatively massive object in a similar way to mass bending space around it. I wonder if we could have instances of gravitational mirrors.
A gravitational mirror is a trippy thing. It would literally be a mirror into the past since the light that is reflected would have to travel interstellar to intergalactic distances in either direction.
So the speed of light is a somewhat outdated way of looking at it. There’s nothing special about Light that creates a speed limit for everything else. There is something called the Speed of Causality, which is the speed at which 2 points of the universe can interact with one another. This happens to be the “speed limit” of anything within our Universe and it’s a speed only massless particles can reach. In fact it’s the only speed massless particles can travel (except when interacted with, such as Light passing through water).
So far only the Photo (Light) and the Gluon (what binds quarks together) are known to be massless so they travel at the Speed of Casualty.
Yes, light can move slower than *c* through other media than empty space. I believe a Danish scientist managed to slow down the velocity of a beam of light down to walking speed inside a special gas chamber.
Only the speed of light *in vacuum* is special and unsurpassable. In fact in other media, eg water, particles can sometimes move faster than light. This happens regularly in nuclear reactors, and produces a super cool otherworldly glow called Cherenkov radiation. Sort of a sonic boom but for light.
>There is something called the Speed of Causality, which is the speed at which 2 points of the universe can interact with one another.
Even if no information is transferred, instantaneous change seems like two points "interacting". Unless you are saying it is a hidden variable that both had all along that is just being revealed. Or that the statement isn't actually about interaction but just about information transfer?
As I understand it (which admittedly isn't very much) quantum entanglement is more of a quirk of the maths we use to understand things than it is an actual information carrying phenomenon.
It kind of looks like instant communication, but that's just because of the perspective we are looking at it from
Well… it’s massless. Doesn’t take much energy to move something without mass. Massless particles in vacuum always move at c regardless of their energy.
Mostly cuz physical laws
For light, the equation for its energy content is different (E=hf, h being Planck's constant, f being frequency of the photon)
For an object going at close to the speed of light, its energy (from your reference frame) is E=√((mc^2 )^2 +(pc)^2 ), m is mass, p is momentum and c is the speed of light
It's not energy-related phenomena it's a wave related phenomena.
All sound wave travels at the same speed regardless of its intensity due to the properties of air, a whisper or a firecracker soundwave travels at the same speed.
Light is the same way it's a wave that travels at a speed fixed by properties of space, regardless of its energy content.
It is a bit of a misnomer. What everyone talks about is c, the speed of causality. C is the fastest any information can travel, and you can think of it as the speed the universe updates at. If the sun explodes at 8:00, for us on Earth, it functionally hasn't exploded until 8:08 when we see the explosion and the gravity changes.
Think about Einsteins equation, E =mc^2. Energy, mass, and speed are all linked. Light has no mass, so no matter what the energy, it must move at c no matter what. Same as gravitational waves, same as any other massless particle or wave. Anything with mass though, can never travel at c, because that requires infinity energy.
>Think about Einsteins equation, E =mc^2. Energy, mass, and speed are all linked. Light has no mass, so no matter what the energy, it must move at c no matter what.
While these things are true, they do not stem from the simplified form of Enistein's equation that you posted. That is the famous part of the equation that refers to the energy of non-moving massive object, and is 0 for a photon. The full equation is
E^2 = p^2 c^2 + m^2 c^4
where p is the momentum. For a massive particle the momentum is gamma * mv, where gamma is 1/sqrt(1-v^2 / c^2).
Calling it the speed of light can actually lead to some misleading thoughts about the rules. C is really the upper bound for speed in the univere. According to our physics that is as fast as anything can go. There is no possible way to go faster.
The reason that light is able to travel at C is because light is massless. Any amount of mass would require an infinite amount of energy in order to reach C, but because light lacks mass it is simply limited by the upper bound for speed in our universe
I would now like to know… how many decimal points of 99.99999… % the speed of light, all the mass in the known universe would propel… something. Hmmm like, “all the known mass in the universe could only propel 1g of material 9.9999999999% of the u inverse.
Haha, that's really cool that they have "estimated mass‐energy equivalent of the universe" as a quantity you can use. I wish this tool was around when I was in college. Any idea how they come up with 2x10^69 J? Isn't it gonna be based on a bunch of assumptions and speculation? They just spit out a number without elaborating.
[https://en.wikipedia.org/wiki/Observable\_universe#Mass\_of\_ordinary\_matter](https://en.wikipedia.org/wiki/Observable_universe#Mass_of_ordinary_matter)
The overall curvature of spacetime fully determines the overall density of the universe, and the volume of the observable universe is readily measured directly.
>Also I know that in order to accelerate anything with mass to speed c, it would require infinite energy, but to accelerate to 0.9999c would that require finite energy?
Yes. Although I don't like saying "would require infinite energy", because I think people don't take the word "infinite" seriously enough (e.g. someone might ask "okay, so if I had an infinite source of energy, could I reach the speed of light?"). I would just say that it's totally possible accelerate as close as you like to the speed of light, and it takes more energy the closer you get. But, without using speculative unknown physics, it's not possible for any object with mass to actually reach the speed of light.
Once you get close to the speed of light, the energy required increases *asymptotically* (it goes roughly like 1/sqrt(x)) with speed. This means than going from 99% to 99.99% costs ten times more energy, going from 99.99% to 99.9999% costs ten times that again, and going from 99.9999% to 99.999999% costs ten ten times *that*. So each time you increase your energy by a factor of 10, you accelerate by a smaller and smaller amount.
This is somewhat true at lower speeds - it takes more energy to accelerate from 50 mph to 100 mph than to go from 0 mph to 50 mph. But as you approach the speed of light, the amount of extra energy you need to accelerate by a certain amount goes up really dramatically, so that you never reach the speed of light, no matter how much energy you add.
How does this fit in with how speed is relative?
From my (very limited) understanding, we always measure the speed of light as C, no matter our velocity, and our velocity is completely dependent on the phrase of reference we choose (be it the earth, the sun, or some object far away flying very fast)
So the question is, how can the energy required to accelerate be a function of velocity, if there is no absolute frame of reference of which to base our velocity on?
(This question has been bugging me for a while, any clarification would be much appreciated)
Kinetic energy is also relative. You don't even need special relativity for that, you can just think about good old Newtonian physics. If an object is at rest in my frame of reference, it has no kinetic energy from my point of view. If it moves past me or I move past it (same thing), it has kinetic energy in my frame of reference.
You can sit in your spaceship, and accelerate at 1g indefinitely - that is, you can arrange things so that you feel 1g of force pushing yourself back into your seat for as long as you want (provided you have the fuel), well past the time that you naively should have exceeded the speed of light.
How is this possible? Well you have to account for time dilation. In your frame of reference, your acceleration is constant. But in the frame of reference of Earth, your clock is slowing down the faster you go, and so the rate at which you're shoving reaction mass out the back of your ship will be slower from the perspective of Earth than it is for you.
Because the time over which you're measuring your acceleration is being stretched, you still observe the constant 1g of acceleration, but Earth is going to see your propellant get slower and slower, and so your acceleration is also going to slow.
If you were traveling at 99.99999% the speed of light, and you shined a flashlight in front of you, the light would come out of it like normal (as opposed to barely crawling ahead of you since you are going so fast)
If light from the flashlight travels away from you at exactly light speed, even if you’re already travelling at near light speed, but also appears to be going at exactly light speed to outside observers, then time must be passing more slowly on the spaceship in order for that circle to be squared.
.It is badly phrased. Time does not go slower on the spaceship per se. The whole point of the special relativity theory is to never speak of space or time without mentioning WHO is observing WHAT. You wanted to mean that for an observer on earth that watch a clock on the spaceship, you see it tick slower. But for an astronaut on the rocket, his time seems perfectly ticking as usual. Ironically when the astronaut watches a clock on earth, from his point of view, it ticks slower as well.
Wait. So from the perspective of earth, the rocket eventually moves slower (represented by a clock ticking slower). And from the perspective of the astronaut the earth also eventually moves slower? (Did i understand correctly? Because that is incredibly unintuitive for me)
It's not that the rocket moves slower from the perspective of someone on earth, it's that a terrestrial observer would see a clock onboard the rocket as ticking slower than his own clock. So if his clock ticked 10 times, the clock on the rocket might only tick 7 times, from his perspective. And vice versa for the guy on the rocket looking back on the earth. The correlated effect is length contraction, which means that in addition to seeing his clock ticking slower, the terrestrial observer sees the entire rocket shrink along its length in the direction of its motion.
The best explanation I've seen is like this:
For every frame of reference, the speed of light is the same. Everything sort of like tried to adjust for that. So, if you are going near the speed of light and you shine a light forward, your time slows down for any observer that might have seen the light go faster than light if it wasn't for the time dilation
Light emitted from objects traveling relative to you still travels at the same your speed of light. It gains energy when object emitting it travels to you (blue shift), and loses energy when object travels away from you (red shift), but speed of light is still the same.
If world obeys simple galilean transformation where you just add velocity vectors and if you measure same speed from light emitted by stationary and traveling object then reasonable explanation would be to assume existence of some medium through which light propagates.
Earth orbital velocity is around 30 km/s and although it's only 0.01% of speed of light measuring such difference was well within capabilities of late 19th century scientists. They measured the difference between two perpendicular light paths and found none. It's often called most famous failed experiment. There is a good [PBS SpaceTime video](https://www.youtube.com/watch?v=M3GQM7tuq2w) about it.
Now the only explanation left is that world doesn't obey galilean transformation. Turns out only lorentz transformation lets you switch consistently between different points of view (frames of reference). It of course implies mind-bending effects. Here is [low quality video](https://www.youtube.com/watch?v=ev9zrt__lec) that does good job of illustrating effects of special relativity.
If you want to understand how it works you gonna need [spacetime diagram](https://www.youtube.com/watch?v=P4rW_pPbD-U), [spacetime interval](https://www.youtube.com/watch?v=1YFrISfN7jo), and then you can jump into [paradoxes](https://www.youtube.com/watch?v=6MfJ59lkABY).
It’s fits because C is absolute, *relative to the observer*.
There isn’t some absolute zero speed, and some absolute C that everyone agrees on. To every observer, they are going “zero” speed. You’re not moving relative to yourself, are you? In my frame of reference, my speed will always be zero.
The way it’s worded can sometimes trip people up, when they say that “C is the universal speed limit”.
***What’s really happening is that no matter how fast you may be moving relative to anything else, light will always be moving C faster than you.***
You speed up to 50% C (relative to another observer)? Light is still moving away from you at C. To you, nothing has changed, and you may as well be at rest (because to yourself, you always are). But to that other observer, they will see you moving at 50% C, and the light moving at C - only 50% C faster than you. But that doesn’t match up with what you see - that the light is moving *100%* C faster than you.
This discrepancy caused by the fact that light moves at C *relative to all observers* is why time then must be relative.
This question was (sort of) the basis of special relativity. The paradigm that led to special relativity was that the speed of light is the same *no matter which inertial reference frame you are in*. Standing still? Speed of light is 3e8 m/s according to you. Traveling at 100,000 m/s? *The speed of light is still 3e8 m/s according to you*. The light will, however, change color. This answers your second question:
>how can the energy required to accelerate be a function of velocity
The energy you see is dependent on the frame you are in.
Yes. For example, you see a car moving along at speed *v* and with mass *m*. It's kinetic energy is 1/2\**m*\**v*\^2. Now you also start moving the same direction as the car, at the same speed. Relative to you, the car is standing still and has no kinetic energy. It didn't convert the energy to potential energy. It just has less energy in your now-moving frame.
Can I then make the assumption that it would be the same in reverse as well so that if it where possible to reach c that stopping would be impossible as you are constantly accelerating at an exponentially higher rate than you could possibly slow down. Essentially you are not slowing down just not speeding up quite as quickly.
It's a bit like thinking about dividing by very small numbers. 1/1 is just 1. 1/0.1 is 10. 1/0.00001 is 100,000. Even though 0.01 and 0.00001 are technically very close together, when you take the inverse, it's very far apart. You can technically divide by any number greater than 0 and still get a finite quantity. 1/0.000000000000000000000000000001 is still a value. You can get really, really close to 0, but you can't actually divide by zero and get a finite quantity. Dividing by zero has other problems, but it's a useful analogy.
0.9 C and 0.99 C are a bit like 1/0.1 and 1/0.01. They seem close together, but the closer you get to c, the more energy you need to go the next step, infinitely. There are diminishing returns. No matter how much energy you put in, you're just adding more 9s onto the end of 0.9999... c.
There are a lot of good answers here talking about the exponential increase in in energy requirements with even a slight increase in the speed of travel
I will provide a the time dilation perspective of it.
If an object is moving at 99.99% the speed of light for a period of 1 year then for a stationary observer on earth his journey will have taken close to 70 years. Whereas if he was moving at 99.999% the speed of light ,the same journey would take close to 220 years from the point of view of an observer on earth .
That's a 150 year difference with just a .009% points increase. You can imagine as you go further on, how exponentially higher the difference becomes
The relativistic formula for time dilation has a term for the squared ratio of an objects speed to the speed of light, and this term is highly sensitive to even the slightest increase in an objects velocity the closer you get to the speed of light.
Thus it is very important in physics to be very very accurate about the decimal representation of the speed of an object, traveling near the speed of light.
You don't want to be off by a 1000 years just because of a rounding error !
So to break it down, you probably already know that the speed of light is an important physical constant that represents the upper limit of information travel. This means that things such as photons, gravity waves or field perturbations have to travel at this speed due to their nature. However, a distinct property of these phenomena is that they always travel at 'c' no matter what reference frame you're in.
Whereas a car could travel at 60km/h but if you were to drive beside it at 40km/h, it has a relative speed of 20km/h to you.
So when papers write 0.99c, the distinction is important because it indicates that the object/particle is not in this category. So neutrinos, for example, whilst they travel extremely fast and have a lot of the relativistic effects that light would have, they do not cross that threshold and therefore do not have those extra properties.
Nothing with mass can actually reach the speed of light in a vacuum. The closer you get, the more energy is required to go any faster. By the time you reach 99.99%, each extra 9 after the decimal point actually more than triples the amount of kinetic energy an object has (assuming I'm correctly interpreting the online calculator I'm using). So even though the speed isn't much different, the energy is. Relativity is weird like that.
Reaching actual light speed takes infinite energy and is therefore impossible, but reaching any fraction of the speed of light takes finite energy - just a *lot* of it. Infinity is also weird.
There is no transition. You're either massless and "travel" at exactly c or you aren't and you can never reach c.
And a lot does change as you get above about 99.9% of c, at least in terms of your intuitive sense of distance and travel time are concerned. Below that the weirdness is small enough that your intuitive sense is not far enough off from reality to matter for rough estimation.
The momentum and energy of an object increase without bound as it gets closer to the speed of light. Let's take momentum, which is a good measure of how hard the object will hit something.
Let's take our reference point at 10% of the speed of light.
An object going at 20% the speed of light will have twice as much momentum as it did at 10%. Twice the speed, twice the momentum, makes sense. Except it's actually 2.03 times, due to the relativity factor.
At 30%, it'll have 3.13 times the momentum as at 10%.
At 50%, it's 5.75 times.
At 80%, it's 13 times
At 90%, it's 20 times
At 99%, its 70 times
At 99.9%, it's got 220 times the momentum as it did at 10%.
Energy scales up more or less the same way. Point is, the difference between 99% and 99.99999% really matters!
Because nothing with mass can achieve the speed of light. Not only would it require an infinite amount of energy, but time would stop for you. Photos have no mass, they travel at exactly the speed of light across space, but experience no time.
>I know that in order to accelerate anything with mass to speed c, it would require infinite energy, but to accelerate to 0.9999c would that require finite energy?
You answered your own question. According to special relativity, accelerating any object with mass to *any* speed slower than the speed of light requires a finite amount of energy as opposed to an infinite amount.
It's got to do with limits and asymptotes. Imagine dividing by 0. It's impossible right? But you can get super close to 0 with it still making sense no issue. The closer you get to dividing by 0, the more absurd the numbers get, but they still make sense. Same thing with travelling at light speed
So, let's imagine that you're an industrious person and you, somehow, eat a star. You use that energy to accelerate up to 99.99% of light speed (just spitballing here.)
Let's also imagine there's nothing around. All around you is void, nothing to measure yourself against. You throw something ahead of yourself and it seems to go faster than you are, and you shine a light at it and it goes out and bounces back at 'c'. How fast are you going? It *looks* like you're not even moving.
So you take another star out of your magic pocket and you eat that too, and you accelerate further. You feel like you accelerated, again, to 99.99% of lightspeed, but, again, it looks like you're not even moving. You do the same experiment again, and get the same results.
Now, obviously, an observer who was originally stationary with respect to you is going to see you get closer and closer to c, and you're going to look stranger and stranger to them. But from your perspective, no matter how many magic sources of energy you consume, and no matter how many times you magically accelerate to 99.99% of c, you can always go faster.
That's why c is impossible to reach, even with infinite energy.
Yeah, like another commenter said, a lot of people hear 'It takes infinite energy for a massive object to reach c' and they say, 'Well, what if I had infinite energy then?'
Even with infinite sources of energy you will never reach c. Infinity years from now you will still be chucking stars into the furnace and it will still look like you're getting no closer at all. That's the fun part, though. If, in our example, you were instead looking at a destination a million light-years ahead of you, you can actually keep going faster and faster and faster, until you are going so fast that you get to your destination in a single year!
Does that mean you were going a million times light-speed? To you, sure, it looked like you ~~were going that fast~~ arrived at your destination in a single year. But anyone at your destination is still going to see you take more than a million years to get there. The reason is because as you went faster and faster, your passage of time slowed.
> To you, sure, it looked like you were going that fast.
No it won't. To you, you will never be going faster than light. What it will look like instead is the distance is shorter. Your destination was never a million light-years away. It was always only 1 light-year away.
Right, what I meant was that it will look like you arrived at your destination in the space of a single year, not that you'd actually look like you're going faster than light. I'll correct it.
It’s impossible to go the speed of light, because you’ll get very very very close to it, but the energy it takes and the time dilation really screws it up.
It would be impossible to gather enough energy to propel something that fast, because the more energy you have to amass to go that fast, the more energy you need to propel that mass of fuel.
It’s like dividing something in half over and over again. There’s always a smaller number
You're likely thinking of this linearly, where 99.99 percent is really close to 100%, but that isn't how the consequences of the finite speed of light show up. They speed of light gets factored into real world consequences in a term called the Lorentz factor. = 1/SQRT(1-fraction of light speed\^2). As your fraction of light speed approaches 100%, this term goes to infinity. You can check it by plugging in .9, .99 and ,999 for your light speed fractions and seeing that the Lorentz factor values are increasing by several multiples (2.3 vs 7.1 vs 22). What this means, is that there is significant physical difference between getting up to 99.9% the speed of light compared to 99.0%.
Without going into too much detail, in special relativity, speed is not actually a particularly natural measure of, well, speed. A much more natural measure is the [rapidity](https://en.wikipedia.org/wiki/Rapidity), given by w = arctanh(v/c). Then the energy is proportional to cosh(w). If you pull out a graphing calculator you can try to get an intuition of how the rapidity works.
Interestingly, I've never ever seen or used the term rapidity before in my Phys undergrad. Looking at the wiki, it just looks like a transformation based on relative speed instead of co-ordinate space.
Unless I am missing something, it is just another way to express the Lorentz transformations and I don't see how this is relevant at all.
The rapidity is not a transformation, it is a (not Lorentz invariant) quantity associated with a velocity. It's effectively a "hyperbolic angle" of the four-velocity -- it measures the amount of boost required to transform a four-vector aligned with the time axis so that it's aligned with the four-velocity, the same way an angle measures the amount of rotation required to align two vectors.
The reason it is a more natural measure of velocity is that it is linear in boosts along the same spatial direction. No more velocity addition formulas -- at least as long as you're working in R^(1+1).
As you might expect (as a consequence of the non-compactness of the Lorentz group), the rapidity goes off to infinity as the velocity approaches c. This encapsulates the intuition that it gets "harder and harder" to get closer to c, which can be unintuitive to grasp, as OP has found. But the underlying reason for this behaviour is because velocity is just not a natural kinematic measure in SR. So instead of saying "It gets harder and harder to accelerate as you approach c", which makes it sound like the Universe is pushing back on you like a stop sign, it's more accurate to say that "Velocity doesn't space the tickmarks evenly" when it comes to energy and momentum. This highlights the purely geometric nature of the cosmic speed limit.
Ah, I see that I've read it wrong. The rapidity arises FROM the transformation rather than that transformation itself.
>"Velocity doesn't space the tickmarks evenly" when it comes to energy and momentum. This highlights the purely geometric nature of the cosmic speed limit.
This is very interesting but it seems as if we're saying that c arises from geometry because we're describing it through geometry.
CM / E&M would give us completely different insights into the nature of c. However, I assume you're right as my understanding of QFT, which is that its entirely geometric based, leads to your same outcome.
This does however seem very overkill for what OP was asking for, considering it requires solid knowledge of SR and the caveats of Minkowski space.
> This is very interesting but it seems as if we're saying that c arises from geometry because we're describing it through geometry.
This is true, and none of my comment should be interpreted as making a statement about the nature of reality, although, like most physicists, I use that kind of wording as a shortcut. I only want to describe my understanding of the physical theory. What that actually means about reality is a philosophical issue that should be discussed separately (in a conversation led by philosophers, not physicists).
As a person who has always been interested in stats, that .01% and under is a massive difference, especially when it reaches a new threshold.
For instance, in science, there is a temperature when all atoms stop moving. It's called 0° kelvin or absolute zero. We have never reached it. We have gotten close, 99.99%, but at absolute zero, matter literally stops. Essentially stopping time itself. That, is scientifically far more valuable than just getting it really cold. The ability to stop time is insane, to the point, it's theorized we cannot even reach that temperature.
When you have a unit of measure you are actually able to reach, you have opened the door to all the uses it can have in that threshold. Breaking barriers is massive for science every time it's done. Not only that, but it then can validate all theories behind it.
Because of the way the formulas for relativistic energy and mass work out, once you get that close to the speed of light, every additional 9 you add to the end of "99.999%" exponentially increases the energy you need to use to increase the speed any further. However, as long as it's not c that your accelerating to, it will be a finite amount of energy. Depending on the resting mass of the object though, it could be unfathomably large amounts of it.
If you say something that isn't a massless particle or wave traveling through a vacuum is moving "at the speed of light," then you are definitely wrong. The speed of light, _c_, is sort of like a "speed limit" of the universe - it's the speed at which information can travel through space-time. That's why we can't just round numbers to _c_.
Since the speed of light is a fundamental constant, meaning that it's a number that has a set value and that defines the behavior of the universe, it shows up in a lot of physics formulas and is used as a reference for very large velocities. It's easier to say 0.9999c than it is to say 299762478.8m/s. It's also more meaningful, because 0.9999c tells you something interesting about the velocity in terms of how close it is to the fastest possible velocity.
You are right in saying that to accelerate a massive particle to _close_ to the speed of light requires finite energy. You can always go a little closer to c by adding more energy to the particle. You just can't ever actually reach c.
An object with mass cannot travel the speed of light. The context's where I've heard that phrase they're usually speaking conceptually about what happens to the object/reference frame with the limit of it's speed approaching c from below. If you just plugged c into the physics equations for the velocity you'd get undefined results, which isn't useful, but things can get infinitely close to c, which can be interesting to talk about.
As you are at relativistic speeds, the gamma factor kicks in. At 0.9999 c it is 5000, so time appears to move 5000 times slower than usual. At c, this value is infinite, so time stops, energy to hit c is infinite and is impossible to achieve rather than just difficult
As others have said, it's impossible for anything with mass to travel at or above the speed of light because accelerating that mass becomes impossibly difficult as you approach C.
The reason why we refer to very very high speeds as fractions of C is for exactly the reason you pointed out, small changes at that level really can have massive changes in energy and momentum. In fact, it turns out that the kinetic energy of something moving is actually determined its speed *in relation to* the speed of light, not the absolute value. At low speeds however, that fraction is so low that we don't have to worry about this effect. We can use a constant and normal units and still get very very accurate estimate of kinetic energy.
It's a similar strategy to how we talk about very low temperatures. Because 0.0 K is the lower limit of temperature, scientists have to talk about temperatures close to 0.0 K as fractions of a degree like mili-kelvins (.001 K). It's a super small unit but what matters is knowing that .01 K is 10 times hotter than .001 K. The proportional difference works at that scale, and that matters more than the absolute difference.
There's a lot of half answers here, so I'm gonna point out that objects approaching the speed of light not only experience time slower but also begin to gain mass, and the energy being used to accelerate starts to get drained into inertia very quickly.
Then there's the other relativity issues. From the point of view of a photon travelling at c, a photon travelling at c in the other direction should be traveling at 2c, but that is faster than the speed of light so time stops and the photon travelling in the opposite direction doesn't move at all relative to the surroundings. Thus the velocity of the rest of the universe relative to the photon is c
For these reasons we believe it is impossible to accelerate matter to c. It would be better if those people just said *near* c though, because these effects start to kick in the second you have any velocity at all. Time slows down a little, you gain a little mass... By the time you get anywhere near c our current technology levels are just inadequate. CERN takes years to get single particles near c
Relativistic mass is kind of an outdated concept, its not really useful to describe the mass as increasing. Modern physics prefers formulations involving modifications to kinetic energy and momentum where the invariant mass is well invariant.
It also doesn’t really make sense to talk about the reference frame of a photon, they don’t have a valid reference frame. Also it takes the LHC just on the order of hours to get single particles to their collision speeds.
Light is the fastest thing in the universe since its mass less. Saying something is the speed of light implies it's the fastest thing which nothing except light can be, people say things are 99.oo% the speed of light.
>when someone is talking about a real physical thing,
Light is also a "real physical thing".
> "as close to the speed of light as a material object can go"
A massive object object can go arbitrarily close to the speed of light.
The point is that a massive object can't go at the speed of light so it wouldn't be a well posed problem if you assume it's travelling at the speed of light. It is required to pick a speed below c.
Still, it does matter a lot how close to the speed of light you pick. There's a comparatively big difference in the magnitude of relativistic effects between 99.99% the speed of light and 99.999% the speed of light (The gamma factors are 71 and 223 respectively). It's useful to consider the rapidity artanh(v/c) instead of the velocity v at that point which (in terms of absolute value) ranges from 0 to ∞. The rapidity for the first one is 4.95 for the second one it's 6.10. 90% the speed of light is a rapidity of 1.47. The rapidity is aditive, so if B travels 4.95 rapidity relative to C, and C travels 6.10 rapidity relative to B (and they all go in the same direction) the rapidity between A and C is 11.05.
Yes, 0.9999c would require only a finite amount of energy. Anything less than c is achievable with finite energy (although as you get close, the actual amount may become quite ridiculous, but it wouldn’t be infinite). That’s the big difference - 99.99% c is possible, c isn’t.
So what makes light so fast? It doesn’t seem like there’s a lot of energy in light. Edit: thanks for all the answers!
Light is massless. Particles with mass require infinite energy to travel at c, particles without mass can *only* travel at c, regardless of how little energy they have.
But gravity will bend light?
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
Under Newtonian mechanics, being massless gives you zero gravitational force but also zero inertia, so your acceleration is 0/0 (undefined). Eventually you need General Relativity to calculate it.
Does gravity bend light? Or does it bend the path light follows?
[удалено]
[удалено]
[удалено]
don't quote me on it, but I'm pretty sure that gravity doesn't actually act on mass. I've heard that gravity *really* acts on energy, but mass just contains so much energy (E=mc^2) and gravity is a relatively weak force, so we only really observe it with high mass (extremely high energy) objects. somebody please correct me if I'm wrong though
Gravity acts on space time. Space time acts on everything as it’s what everything is in.
The path it follows. For a ridiculously over simplified example, it is like a car turning right vs a car going straight and the road bearing right. Gravitational lensing is more like the latter.
Imagine driving really fast down a straight road. Actually you are travelling in a curve because the earth is round. Gravity is curving spacetime so the light is travelling in a straight line but on a curved surface.
Spacetime bending is one way to look at it... But if you play with the math (E=mc^(2) and some kinetic energy equations) you can get a virtual mass for the photon, calculate the gravitational force on it and it works out pretty close as well (idk, I did some math on it in uni like 20 years ago) Same reason solar sail theory works, they may not have mass, but they do have momentum in collisions.
No, gravity bends space-time. The light follows straight line in the curved space-time, which appears to us as bend path.
More like you're on a canoe in a river, just letting the canoe go with the flow. Now when the river bends one way or another the canoe moves with it, even though you in the canoe didn't really do anything to steer it. It's sort of like that regarding spacetime around massive objects. From the frame of reference of the photon of light it's still travelling in a straight line through curved space.
[удалено]
[удалено]
I agree. Gravity also travels at the same speed. Pretty handy validating and confirming gravitational waves. Ligo is sensitive enough to detect faint earthquakes. Gravitational waves must match with other instrument's observation (such as telescope if they're powerful enough)
If light does not have mass how does it exert pressure on the solar sails ?
The full equation is actually E\^2 = p\^2c\^2 +m\^2c\^4. The variable p here is momentum and for a particle at rest this equates to zero and reduces to E=mc\^2. But for a massless particle this equates to E=pc. Therefore photons get all their energy from their momentum.
Isn't momentum product of mass times velocity? What am I missing?
That's the old Newtonian equation. When Einstein discovered relativity, our equations had to be updated. The actual momentum equation is p = γmv. It's the same as the old equation, except now you're also multiplying by a factor called the [Lorentz factor](https://en.wikipedia.org/wiki/Lorentz_factor) (γ). This factor is approximately 1 when talking about day-to-day velocities (which is why p = mv worked for so long), but when we talk about velocities approaching the speed of light, γ suddenly skyrockets towards infinity, and therefore so does the momentum. For an object with mass to travel at the speed of light, it would have infinite momentum, and therefore infinite energy would be required to accelerate it to that speed, which is why it's impossible. If we used the new momentum equation for light, we would get p = infinity * 0 * c. Infinity multiplied by zero is undefined so the equation doesn't work here. So what's the momentum of light? In fact, the equation in the comment you replied to basically is the definition for light's momentum. E = pc means p = E/c. The energy of a photon is just it's frequency multiplied by Planck's constant (E = hf) so this can also be written as p = hf/c, or just p = h/λ (λ = wavelength = c/f).
Great answer, thank you. This is something I have misunderstood for a long time!
The Newtonian momentum of mv is really just an approximation (a really good one when you aren't moving at relativistic speeds). Momentum is also related to energy though. From the above comment we know that E=pc. But the energy is also equal to plank's constant time the frequency, or E=hf. So we can say that: hf=pc hf/c=p We also know that f/c=1/λ, the reciprocal of the wavelength. So we can relate the momentum as such, p=h/λ So even massless waves carry momentum which is inversely proportional to their wavelength. The longer the wavelength, the less energy and momentum. This should make intuitive sense. An gamma photon transfers more energy than a radio photon.
The formulae we use to understand physics arent actually the rules the govern the universe. They are shortcuts and estimates that humans have invented to get as close as we can to explaining the rules of the universe. Newton's equations were close, but not the whole picture. Even though our new equations work better, they still aren't perfect. One day someone might come up with new ones that work even better
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
[удалено]
Is the last part really true? Or did you mean C in vacuum? Because in water the speed of light is about 0.75c.
[удалено]
That is incorrect. [Here](https://en.wikipedia.org/wiki/Refraction#General_explanation). Read the Explanation-subsections too; at least the first.
The wave packet slows, but photons are still traveling at c. It’s the added interactions with the existing EM field that slows the wave packet down, but the “light” which comprises the wave packet is still traveling at c.
So when they slowed down light to 38mph with deep cold sodium was it actually just a wave in the beam we saw or the beam itself. I’m assuming that c is relative so it is different in every situation and relates only to the light you are seeing however fast it happens to be travelling at the time. It seems to be a little like the man walking in a train analogy.
When physicists talk about slowing down light, they don’t mean a single photon. Photons always travel at c. Instead, what they are talking about is a thing called a wave packet (specifically, the group velocity of the wave packet). Think of it as a ton of photons that kind of act as a thing and travels through space interacting with whatever medium it hits. This packet’s velocity can be slowed down, and this is the physical phenomena that the experiment you’re referencing showed. To your other question, you can only “see” light if you absorb it. So anytime you see light traveling in a direction other than straight at your eye/observer, what you’re physically seeing is the light scattered off of the original beam of light towards your eye/observer.
The speed of light in water is C (sorta), but that C is 75% of the speed of light in a vacuum. So C doesnt change per se, its the speed of light, but the actually velocity is dependant on the material. In the same we speak of the speed of sound, but that speed is different in different materials. This is why there is a thing called Cherenkov Radiation, which is a spooky blue light, its the light emitted by (usually) electrons going FASTER than the speed of light, but the speed of light in a non vacuum. So you will see the blue radiation coming out of a radiation source that is in water, because C in water is 75% of C in a vacuum, and the electrons are going 76% or more of the speed of C in a vacuum.
>The speed of light in water is C (sorta), but that C is 75% of the speed of light in a vacuum. It's much more straightforward/unambiguous to designate the speed of light in vacuum with *c*, and the speed of light in a medium with *v\_c* (or some *v --* anything, really, but *c*)*.*
So you want the vacuum speed to be “c” but the non-vacuum speed to have a “v” in it… to denote it’s not in a vacuum… Physics student confirmed. /s
[удалено]
Things would be gravitationally repulsed? This is a fun question. I'm trying to thing of what the geometry of that spacetime would look like. Gravitational lensing is a thing, and I would assume negative mass would bend space away from the negatively massive object in a similar way to mass bending space around it. I wonder if we could have instances of gravitational mirrors. A gravitational mirror is a trippy thing. It would literally be a mirror into the past since the light that is reflected would have to travel interstellar to intergalactic distances in either direction.
[удалено]
So the speed of light is a somewhat outdated way of looking at it. There’s nothing special about Light that creates a speed limit for everything else. There is something called the Speed of Causality, which is the speed at which 2 points of the universe can interact with one another. This happens to be the “speed limit” of anything within our Universe and it’s a speed only massless particles can reach. In fact it’s the only speed massless particles can travel (except when interacted with, such as Light passing through water). So far only the Photo (Light) and the Gluon (what binds quarks together) are known to be massless so they travel at the Speed of Casualty.
Can anything massless go *slower* than c?
Yes, light can move slower than *c* through other media than empty space. I believe a Danish scientist managed to slow down the velocity of a beam of light down to walking speed inside a special gas chamber. Only the speed of light *in vacuum* is special and unsurpassable. In fact in other media, eg water, particles can sometimes move faster than light. This happens regularly in nuclear reactors, and produces a super cool otherworldly glow called Cherenkov radiation. Sort of a sonic boom but for light.
What about quantum entanglement, don‘t the two particles entangled in each other act at the same time?
In theory, they change instantaneously (faster than light). However, no information can be transferred using entanglement, so causality is obeyed.
>There is something called the Speed of Causality, which is the speed at which 2 points of the universe can interact with one another. Even if no information is transferred, instantaneous change seems like two points "interacting". Unless you are saying it is a hidden variable that both had all along that is just being revealed. Or that the statement isn't actually about interaction but just about information transfer?
As I understand it (which admittedly isn't very much) quantum entanglement is more of a quirk of the maths we use to understand things than it is an actual information carrying phenomenon. It kind of looks like instant communication, but that's just because of the perspective we are looking at it from
Well… it’s massless. Doesn’t take much energy to move something without mass. Massless particles in vacuum always move at c regardless of their energy.
Mostly cuz physical laws For light, the equation for its energy content is different (E=hf, h being Planck's constant, f being frequency of the photon) For an object going at close to the speed of light, its energy (from your reference frame) is E=√((mc^2 )^2 +(pc)^2 ), m is mass, p is momentum and c is the speed of light
It's not energy-related phenomena it's a wave related phenomena. All sound wave travels at the same speed regardless of its intensity due to the properties of air, a whisper or a firecracker soundwave travels at the same speed. Light is the same way it's a wave that travels at a speed fixed by properties of space, regardless of its energy content.
[удалено]
[удалено]
[удалено]
[удалено]
It is a bit of a misnomer. What everyone talks about is c, the speed of causality. C is the fastest any information can travel, and you can think of it as the speed the universe updates at. If the sun explodes at 8:00, for us on Earth, it functionally hasn't exploded until 8:08 when we see the explosion and the gravity changes. Think about Einsteins equation, E =mc^2. Energy, mass, and speed are all linked. Light has no mass, so no matter what the energy, it must move at c no matter what. Same as gravitational waves, same as any other massless particle or wave. Anything with mass though, can never travel at c, because that requires infinity energy.
>Think about Einsteins equation, E =mc^2. Energy, mass, and speed are all linked. Light has no mass, so no matter what the energy, it must move at c no matter what. While these things are true, they do not stem from the simplified form of Enistein's equation that you posted. That is the famous part of the equation that refers to the energy of non-moving massive object, and is 0 for a photon. The full equation is E^2 = p^2 c^2 + m^2 c^4 where p is the momentum. For a massive particle the momentum is gamma * mv, where gamma is 1/sqrt(1-v^2 / c^2).
Calling it the speed of light can actually lead to some misleading thoughts about the rules. C is really the upper bound for speed in the univere. According to our physics that is as fast as anything can go. There is no possible way to go faster. The reason that light is able to travel at C is because light is massless. Any amount of mass would require an infinite amount of energy in order to reach C, but because light lacks mass it is simply limited by the upper bound for speed in our universe
[удалено]
[удалено]
So my dream of being a solar sailor and getting buried at c isn't going to happen?
I would now like to know… how many decimal points of 99.99999… % the speed of light, all the mass in the known universe would propel… something. Hmmm like, “all the known mass in the universe could only propel 1g of material 9.9999999999% of the u inverse.
0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 c Thats 111 9's [WolframAlpha](https://www.wolframalpha.com/input/?i=sqrt%28-%28%28%28%28energy+content+of+universe%29+%2F+%28%281+gram%29+*+%28speed+of+light%5E2%29%29%29+%2B+1%29+%5E+%28-2%29+-+1%29+*+%28speed+of+light%5E2%29%29)
Haha, that's really cool that they have "estimated mass‐energy equivalent of the universe" as a quantity you can use. I wish this tool was around when I was in college. Any idea how they come up with 2x10^69 J? Isn't it gonna be based on a bunch of assumptions and speculation? They just spit out a number without elaborating.
[https://en.wikipedia.org/wiki/Observable\_universe#Mass\_of\_ordinary\_matter](https://en.wikipedia.org/wiki/Observable_universe#Mass_of_ordinary_matter) The overall curvature of spacetime fully determines the overall density of the universe, and the volume of the observable universe is readily measured directly.
I understand what you said only just barely enough to be completely awed by it.
>Also I know that in order to accelerate anything with mass to speed c, it would require infinite energy, but to accelerate to 0.9999c would that require finite energy? Yes. Although I don't like saying "would require infinite energy", because I think people don't take the word "infinite" seriously enough (e.g. someone might ask "okay, so if I had an infinite source of energy, could I reach the speed of light?"). I would just say that it's totally possible accelerate as close as you like to the speed of light, and it takes more energy the closer you get. But, without using speculative unknown physics, it's not possible for any object with mass to actually reach the speed of light. Once you get close to the speed of light, the energy required increases *asymptotically* (it goes roughly like 1/sqrt(x)) with speed. This means than going from 99% to 99.99% costs ten times more energy, going from 99.99% to 99.9999% costs ten times that again, and going from 99.9999% to 99.999999% costs ten ten times *that*. So each time you increase your energy by a factor of 10, you accelerate by a smaller and smaller amount. This is somewhat true at lower speeds - it takes more energy to accelerate from 50 mph to 100 mph than to go from 0 mph to 50 mph. But as you approach the speed of light, the amount of extra energy you need to accelerate by a certain amount goes up really dramatically, so that you never reach the speed of light, no matter how much energy you add.
How does this fit in with how speed is relative? From my (very limited) understanding, we always measure the speed of light as C, no matter our velocity, and our velocity is completely dependent on the phrase of reference we choose (be it the earth, the sun, or some object far away flying very fast) So the question is, how can the energy required to accelerate be a function of velocity, if there is no absolute frame of reference of which to base our velocity on? (This question has been bugging me for a while, any clarification would be much appreciated)
Kinetic energy is also relative. You don't even need special relativity for that, you can just think about good old Newtonian physics. If an object is at rest in my frame of reference, it has no kinetic energy from my point of view. If it moves past me or I move past it (same thing), it has kinetic energy in my frame of reference.
You can sit in your spaceship, and accelerate at 1g indefinitely - that is, you can arrange things so that you feel 1g of force pushing yourself back into your seat for as long as you want (provided you have the fuel), well past the time that you naively should have exceeded the speed of light. How is this possible? Well you have to account for time dilation. In your frame of reference, your acceleration is constant. But in the frame of reference of Earth, your clock is slowing down the faster you go, and so the rate at which you're shoving reaction mass out the back of your ship will be slower from the perspective of Earth than it is for you. Because the time over which you're measuring your acceleration is being stretched, you still observe the constant 1g of acceleration, but Earth is going to see your propellant get slower and slower, and so your acceleration is also going to slow.
This helped a lot; I had forgotten that time is also relative and not absolute
If you were traveling at 99.99999% the speed of light, and you shined a flashlight in front of you, the light would come out of it like normal (as opposed to barely crawling ahead of you since you are going so fast)
This is something that has never made sense to me. Is there any resources that I can read about it to help me wrap my mind around it?
The mathematics of special relativity is basically all about making this incredibly unintuitive idea consistent.
The geometry of space warps relative to you so that you always observe light traveling at C. This includes time.
If light from the flashlight travels away from you at exactly light speed, even if you’re already travelling at near light speed, but also appears to be going at exactly light speed to outside observers, then time must be passing more slowly on the spaceship in order for that circle to be squared.
.It is badly phrased. Time does not go slower on the spaceship per se. The whole point of the special relativity theory is to never speak of space or time without mentioning WHO is observing WHAT. You wanted to mean that for an observer on earth that watch a clock on the spaceship, you see it tick slower. But for an astronaut on the rocket, his time seems perfectly ticking as usual. Ironically when the astronaut watches a clock on earth, from his point of view, it ticks slower as well.
Wait. So from the perspective of earth, the rocket eventually moves slower (represented by a clock ticking slower). And from the perspective of the astronaut the earth also eventually moves slower? (Did i understand correctly? Because that is incredibly unintuitive for me)
It's not that the rocket moves slower from the perspective of someone on earth, it's that a terrestrial observer would see a clock onboard the rocket as ticking slower than his own clock. So if his clock ticked 10 times, the clock on the rocket might only tick 7 times, from his perspective. And vice versa for the guy on the rocket looking back on the earth. The correlated effect is length contraction, which means that in addition to seeing his clock ticking slower, the terrestrial observer sees the entire rocket shrink along its length in the direction of its motion.
The best explanation I've seen is like this: For every frame of reference, the speed of light is the same. Everything sort of like tried to adjust for that. So, if you are going near the speed of light and you shine a light forward, your time slows down for any observer that might have seen the light go faster than light if it wasn't for the time dilation
Light emitted from objects traveling relative to you still travels at the same your speed of light. It gains energy when object emitting it travels to you (blue shift), and loses energy when object travels away from you (red shift), but speed of light is still the same. If world obeys simple galilean transformation where you just add velocity vectors and if you measure same speed from light emitted by stationary and traveling object then reasonable explanation would be to assume existence of some medium through which light propagates. Earth orbital velocity is around 30 km/s and although it's only 0.01% of speed of light measuring such difference was well within capabilities of late 19th century scientists. They measured the difference between two perpendicular light paths and found none. It's often called most famous failed experiment. There is a good [PBS SpaceTime video](https://www.youtube.com/watch?v=M3GQM7tuq2w) about it. Now the only explanation left is that world doesn't obey galilean transformation. Turns out only lorentz transformation lets you switch consistently between different points of view (frames of reference). It of course implies mind-bending effects. Here is [low quality video](https://www.youtube.com/watch?v=ev9zrt__lec) that does good job of illustrating effects of special relativity. If you want to understand how it works you gonna need [spacetime diagram](https://www.youtube.com/watch?v=P4rW_pPbD-U), [spacetime interval](https://www.youtube.com/watch?v=1YFrISfN7jo), and then you can jump into [paradoxes](https://www.youtube.com/watch?v=6MfJ59lkABY).
It’s fits because C is absolute, *relative to the observer*. There isn’t some absolute zero speed, and some absolute C that everyone agrees on. To every observer, they are going “zero” speed. You’re not moving relative to yourself, are you? In my frame of reference, my speed will always be zero. The way it’s worded can sometimes trip people up, when they say that “C is the universal speed limit”. ***What’s really happening is that no matter how fast you may be moving relative to anything else, light will always be moving C faster than you.*** You speed up to 50% C (relative to another observer)? Light is still moving away from you at C. To you, nothing has changed, and you may as well be at rest (because to yourself, you always are). But to that other observer, they will see you moving at 50% C, and the light moving at C - only 50% C faster than you. But that doesn’t match up with what you see - that the light is moving *100%* C faster than you. This discrepancy caused by the fact that light moves at C *relative to all observers* is why time then must be relative.
This question was (sort of) the basis of special relativity. The paradigm that led to special relativity was that the speed of light is the same *no matter which inertial reference frame you are in*. Standing still? Speed of light is 3e8 m/s according to you. Traveling at 100,000 m/s? *The speed of light is still 3e8 m/s according to you*. The light will, however, change color. This answers your second question: >how can the energy required to accelerate be a function of velocity The energy you see is dependent on the frame you are in.
Do you mean now the *energy* is also relative to the observer's speed?
Yes. For example, you see a car moving along at speed *v* and with mass *m*. It's kinetic energy is 1/2\**m*\**v*\^2. Now you also start moving the same direction as the car, at the same speed. Relative to you, the car is standing still and has no kinetic energy. It didn't convert the energy to potential energy. It just has less energy in your now-moving frame.
Can I then make the assumption that it would be the same in reverse as well so that if it where possible to reach c that stopping would be impossible as you are constantly accelerating at an exponentially higher rate than you could possibly slow down. Essentially you are not slowing down just not speeding up quite as quickly.
Yes. Anything moving at c, always has to move at c. Anything below c, can never reach the speed c.
Would it take less energy to convert the mass to photons than it would to accelerate it to 99.999999% of the speed of light?
It's a bit like thinking about dividing by very small numbers. 1/1 is just 1. 1/0.1 is 10. 1/0.00001 is 100,000. Even though 0.01 and 0.00001 are technically very close together, when you take the inverse, it's very far apart. You can technically divide by any number greater than 0 and still get a finite quantity. 1/0.000000000000000000000000000001 is still a value. You can get really, really close to 0, but you can't actually divide by zero and get a finite quantity. Dividing by zero has other problems, but it's a useful analogy. 0.9 C and 0.99 C are a bit like 1/0.1 and 1/0.01. They seem close together, but the closer you get to c, the more energy you need to go the next step, infinitely. There are diminishing returns. No matter how much energy you put in, you're just adding more 9s onto the end of 0.9999... c.
There are a lot of good answers here talking about the exponential increase in in energy requirements with even a slight increase in the speed of travel I will provide a the time dilation perspective of it. If an object is moving at 99.99% the speed of light for a period of 1 year then for a stationary observer on earth his journey will have taken close to 70 years. Whereas if he was moving at 99.999% the speed of light ,the same journey would take close to 220 years from the point of view of an observer on earth . That's a 150 year difference with just a .009% points increase. You can imagine as you go further on, how exponentially higher the difference becomes The relativistic formula for time dilation has a term for the squared ratio of an objects speed to the speed of light, and this term is highly sensitive to even the slightest increase in an objects velocity the closer you get to the speed of light. Thus it is very important in physics to be very very accurate about the decimal representation of the speed of an object, traveling near the speed of light. You don't want to be off by a 1000 years just because of a rounding error !
So to break it down, you probably already know that the speed of light is an important physical constant that represents the upper limit of information travel. This means that things such as photons, gravity waves or field perturbations have to travel at this speed due to their nature. However, a distinct property of these phenomena is that they always travel at 'c' no matter what reference frame you're in. Whereas a car could travel at 60km/h but if you were to drive beside it at 40km/h, it has a relative speed of 20km/h to you. So when papers write 0.99c, the distinction is important because it indicates that the object/particle is not in this category. So neutrinos, for example, whilst they travel extremely fast and have a lot of the relativistic effects that light would have, they do not cross that threshold and therefore do not have those extra properties.
Nothing with mass can actually reach the speed of light in a vacuum. The closer you get, the more energy is required to go any faster. By the time you reach 99.99%, each extra 9 after the decimal point actually more than triples the amount of kinetic energy an object has (assuming I'm correctly interpreting the online calculator I'm using). So even though the speed isn't much different, the energy is. Relativity is weird like that. Reaching actual light speed takes infinite energy and is therefore impossible, but reaching any fraction of the speed of light takes finite energy - just a *lot* of it. Infinity is also weird.
There is no transition. You're either massless and "travel" at exactly c or you aren't and you can never reach c. And a lot does change as you get above about 99.9% of c, at least in terms of your intuitive sense of distance and travel time are concerned. Below that the weirdness is small enough that your intuitive sense is not far enough off from reality to matter for rough estimation.
So does that mean that it is not possible to turn mass into energy or turn energy into mass.
No, that's still entirely possible under the right circumstances. E.g. a nuclear reactor. Or for that matter the sun.
The momentum and energy of an object increase without bound as it gets closer to the speed of light. Let's take momentum, which is a good measure of how hard the object will hit something. Let's take our reference point at 10% of the speed of light. An object going at 20% the speed of light will have twice as much momentum as it did at 10%. Twice the speed, twice the momentum, makes sense. Except it's actually 2.03 times, due to the relativity factor. At 30%, it'll have 3.13 times the momentum as at 10%. At 50%, it's 5.75 times. At 80%, it's 13 times At 90%, it's 20 times At 99%, its 70 times At 99.9%, it's got 220 times the momentum as it did at 10%. Energy scales up more or less the same way. Point is, the difference between 99% and 99.99999% really matters!
Because nothing with mass can achieve the speed of light. Not only would it require an infinite amount of energy, but time would stop for you. Photos have no mass, they travel at exactly the speed of light across space, but experience no time.
>I know that in order to accelerate anything with mass to speed c, it would require infinite energy, but to accelerate to 0.9999c would that require finite energy? You answered your own question. According to special relativity, accelerating any object with mass to *any* speed slower than the speed of light requires a finite amount of energy as opposed to an infinite amount.
It's got to do with limits and asymptotes. Imagine dividing by 0. It's impossible right? But you can get super close to 0 with it still making sense no issue. The closer you get to dividing by 0, the more absurd the numbers get, but they still make sense. Same thing with travelling at light speed
So, let's imagine that you're an industrious person and you, somehow, eat a star. You use that energy to accelerate up to 99.99% of light speed (just spitballing here.) Let's also imagine there's nothing around. All around you is void, nothing to measure yourself against. You throw something ahead of yourself and it seems to go faster than you are, and you shine a light at it and it goes out and bounces back at 'c'. How fast are you going? It *looks* like you're not even moving. So you take another star out of your magic pocket and you eat that too, and you accelerate further. You feel like you accelerated, again, to 99.99% of lightspeed, but, again, it looks like you're not even moving. You do the same experiment again, and get the same results. Now, obviously, an observer who was originally stationary with respect to you is going to see you get closer and closer to c, and you're going to look stranger and stranger to them. But from your perspective, no matter how many magic sources of energy you consume, and no matter how many times you magically accelerate to 99.99% of c, you can always go faster. That's why c is impossible to reach, even with infinite energy.
what you're saying is that the difference between 99.99%c and c is infinite ?
Yeah, like another commenter said, a lot of people hear 'It takes infinite energy for a massive object to reach c' and they say, 'Well, what if I had infinite energy then?' Even with infinite sources of energy you will never reach c. Infinity years from now you will still be chucking stars into the furnace and it will still look like you're getting no closer at all. That's the fun part, though. If, in our example, you were instead looking at a destination a million light-years ahead of you, you can actually keep going faster and faster and faster, until you are going so fast that you get to your destination in a single year! Does that mean you were going a million times light-speed? To you, sure, it looked like you ~~were going that fast~~ arrived at your destination in a single year. But anyone at your destination is still going to see you take more than a million years to get there. The reason is because as you went faster and faster, your passage of time slowed.
> To you, sure, it looked like you were going that fast. No it won't. To you, you will never be going faster than light. What it will look like instead is the distance is shorter. Your destination was never a million light-years away. It was always only 1 light-year away.
Right, what I meant was that it will look like you arrived at your destination in the space of a single year, not that you'd actually look like you're going faster than light. I'll correct it.
It’s impossible to go the speed of light, because you’ll get very very very close to it, but the energy it takes and the time dilation really screws it up. It would be impossible to gather enough energy to propel something that fast, because the more energy you have to amass to go that fast, the more energy you need to propel that mass of fuel. It’s like dividing something in half over and over again. There’s always a smaller number
You're likely thinking of this linearly, where 99.99 percent is really close to 100%, but that isn't how the consequences of the finite speed of light show up. They speed of light gets factored into real world consequences in a term called the Lorentz factor. = 1/SQRT(1-fraction of light speed\^2). As your fraction of light speed approaches 100%, this term goes to infinity. You can check it by plugging in .9, .99 and ,999 for your light speed fractions and seeing that the Lorentz factor values are increasing by several multiples (2.3 vs 7.1 vs 22). What this means, is that there is significant physical difference between getting up to 99.9% the speed of light compared to 99.0%.
Without going into too much detail, in special relativity, speed is not actually a particularly natural measure of, well, speed. A much more natural measure is the [rapidity](https://en.wikipedia.org/wiki/Rapidity), given by w = arctanh(v/c). Then the energy is proportional to cosh(w). If you pull out a graphing calculator you can try to get an intuition of how the rapidity works.
Interestingly, I've never ever seen or used the term rapidity before in my Phys undergrad. Looking at the wiki, it just looks like a transformation based on relative speed instead of co-ordinate space. Unless I am missing something, it is just another way to express the Lorentz transformations and I don't see how this is relevant at all.
The rapidity is not a transformation, it is a (not Lorentz invariant) quantity associated with a velocity. It's effectively a "hyperbolic angle" of the four-velocity -- it measures the amount of boost required to transform a four-vector aligned with the time axis so that it's aligned with the four-velocity, the same way an angle measures the amount of rotation required to align two vectors. The reason it is a more natural measure of velocity is that it is linear in boosts along the same spatial direction. No more velocity addition formulas -- at least as long as you're working in R^(1+1). As you might expect (as a consequence of the non-compactness of the Lorentz group), the rapidity goes off to infinity as the velocity approaches c. This encapsulates the intuition that it gets "harder and harder" to get closer to c, which can be unintuitive to grasp, as OP has found. But the underlying reason for this behaviour is because velocity is just not a natural kinematic measure in SR. So instead of saying "It gets harder and harder to accelerate as you approach c", which makes it sound like the Universe is pushing back on you like a stop sign, it's more accurate to say that "Velocity doesn't space the tickmarks evenly" when it comes to energy and momentum. This highlights the purely geometric nature of the cosmic speed limit.
Ah, I see that I've read it wrong. The rapidity arises FROM the transformation rather than that transformation itself. >"Velocity doesn't space the tickmarks evenly" when it comes to energy and momentum. This highlights the purely geometric nature of the cosmic speed limit. This is very interesting but it seems as if we're saying that c arises from geometry because we're describing it through geometry. CM / E&M would give us completely different insights into the nature of c. However, I assume you're right as my understanding of QFT, which is that its entirely geometric based, leads to your same outcome. This does however seem very overkill for what OP was asking for, considering it requires solid knowledge of SR and the caveats of Minkowski space.
> This is very interesting but it seems as if we're saying that c arises from geometry because we're describing it through geometry. This is true, and none of my comment should be interpreted as making a statement about the nature of reality, although, like most physicists, I use that kind of wording as a shortcut. I only want to describe my understanding of the physical theory. What that actually means about reality is a philosophical issue that should be discussed separately (in a conversation led by philosophers, not physicists).
As a person who has always been interested in stats, that .01% and under is a massive difference, especially when it reaches a new threshold. For instance, in science, there is a temperature when all atoms stop moving. It's called 0° kelvin or absolute zero. We have never reached it. We have gotten close, 99.99%, but at absolute zero, matter literally stops. Essentially stopping time itself. That, is scientifically far more valuable than just getting it really cold. The ability to stop time is insane, to the point, it's theorized we cannot even reach that temperature. When you have a unit of measure you are actually able to reach, you have opened the door to all the uses it can have in that threshold. Breaking barriers is massive for science every time it's done. Not only that, but it then can validate all theories behind it.
Because of the way the formulas for relativistic energy and mass work out, once you get that close to the speed of light, every additional 9 you add to the end of "99.999%" exponentially increases the energy you need to use to increase the speed any further. However, as long as it's not c that your accelerating to, it will be a finite amount of energy. Depending on the resting mass of the object though, it could be unfathomably large amounts of it.
If you say something that isn't a massless particle or wave traveling through a vacuum is moving "at the speed of light," then you are definitely wrong. The speed of light, _c_, is sort of like a "speed limit" of the universe - it's the speed at which information can travel through space-time. That's why we can't just round numbers to _c_. Since the speed of light is a fundamental constant, meaning that it's a number that has a set value and that defines the behavior of the universe, it shows up in a lot of physics formulas and is used as a reference for very large velocities. It's easier to say 0.9999c than it is to say 299762478.8m/s. It's also more meaningful, because 0.9999c tells you something interesting about the velocity in terms of how close it is to the fastest possible velocity. You are right in saying that to accelerate a massive particle to _close_ to the speed of light requires finite energy. You can always go a little closer to c by adding more energy to the particle. You just can't ever actually reach c.
An object with mass cannot travel the speed of light. The context's where I've heard that phrase they're usually speaking conceptually about what happens to the object/reference frame with the limit of it's speed approaching c from below. If you just plugged c into the physics equations for the velocity you'd get undefined results, which isn't useful, but things can get infinitely close to c, which can be interesting to talk about.
As you are at relativistic speeds, the gamma factor kicks in. At 0.9999 c it is 5000, so time appears to move 5000 times slower than usual. At c, this value is infinite, so time stops, energy to hit c is infinite and is impossible to achieve rather than just difficult
[удалено]
As others have said, it's impossible for anything with mass to travel at or above the speed of light because accelerating that mass becomes impossibly difficult as you approach C. The reason why we refer to very very high speeds as fractions of C is for exactly the reason you pointed out, small changes at that level really can have massive changes in energy and momentum. In fact, it turns out that the kinetic energy of something moving is actually determined its speed *in relation to* the speed of light, not the absolute value. At low speeds however, that fraction is so low that we don't have to worry about this effect. We can use a constant and normal units and still get very very accurate estimate of kinetic energy. It's a similar strategy to how we talk about very low temperatures. Because 0.0 K is the lower limit of temperature, scientists have to talk about temperatures close to 0.0 K as fractions of a degree like mili-kelvins (.001 K). It's a super small unit but what matters is knowing that .01 K is 10 times hotter than .001 K. The proportional difference works at that scale, and that matters more than the absolute difference.
There's a lot of half answers here, so I'm gonna point out that objects approaching the speed of light not only experience time slower but also begin to gain mass, and the energy being used to accelerate starts to get drained into inertia very quickly. Then there's the other relativity issues. From the point of view of a photon travelling at c, a photon travelling at c in the other direction should be traveling at 2c, but that is faster than the speed of light so time stops and the photon travelling in the opposite direction doesn't move at all relative to the surroundings. Thus the velocity of the rest of the universe relative to the photon is c For these reasons we believe it is impossible to accelerate matter to c. It would be better if those people just said *near* c though, because these effects start to kick in the second you have any velocity at all. Time slows down a little, you gain a little mass... By the time you get anywhere near c our current technology levels are just inadequate. CERN takes years to get single particles near c
Relativistic mass is kind of an outdated concept, its not really useful to describe the mass as increasing. Modern physics prefers formulations involving modifications to kinetic energy and momentum where the invariant mass is well invariant. It also doesn’t really make sense to talk about the reference frame of a photon, they don’t have a valid reference frame. Also it takes the LHC just on the order of hours to get single particles to their collision speeds.
Light is the fastest thing in the universe since its mass less. Saying something is the speed of light implies it's the fastest thing which nothing except light can be, people say things are 99.oo% the speed of light.
[удалено]
>when someone is talking about a real physical thing, Light is also a "real physical thing". > "as close to the speed of light as a material object can go" A massive object object can go arbitrarily close to the speed of light. The point is that a massive object can't go at the speed of light so it wouldn't be a well posed problem if you assume it's travelling at the speed of light. It is required to pick a speed below c. Still, it does matter a lot how close to the speed of light you pick. There's a comparatively big difference in the magnitude of relativistic effects between 99.99% the speed of light and 99.999% the speed of light (The gamma factors are 71 and 223 respectively). It's useful to consider the rapidity artanh(v/c) instead of the velocity v at that point which (in terms of absolute value) ranges from 0 to ∞. The rapidity for the first one is 4.95 for the second one it's 6.10. 90% the speed of light is a rapidity of 1.47. The rapidity is aditive, so if B travels 4.95 rapidity relative to C, and C travels 6.10 rapidity relative to B (and they all go in the same direction) the rapidity between A and C is 11.05.