## Why does c =c ?

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Taure
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### Why does c =c ?

My question is about the speed of light.

I understand that something with mass cannot reach the speed of light because as it approaches the speed of light its mass will increase, thus meaning it requires more energy to accelerate it, and that you'd need infinite energy therefore to reach c.

Because light has no mass, it can reach the speed of light.

My question is this:

Why does light, which has no mass, only reach the speed it does? What limiting factor prevents it from accelerating faster, given that it has no mass?

doogly
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### Re: Why does c =c ?

The speed of light needs to be the same in all reference frames. That's the essential fact that drives all the other funny business.

Perhaps you may think, why can't I go add some more energy to this light over here? Why can't I add as much as I want? You can! But that just raises its frequency. The speed doesn't change.
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Taure
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### Re: Why does c =c ?

That seems a little unsatisfying... I mean, it tells us what occurs, but not why. There must be a reason why the speed of light is the speed it is, and not a different speed.

I guess we just don't know what it is yet.

Steax
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### Re: Why does c =c ?

Because known physics limits it at so? All electromagnetic radiation travels at that speed, and that speed only. It's embedded into physics, pretty much.

There are theories* that the speed of light has been different in the past, or may change in the future. It appears to simply be one of those magical constants (or, constant at this time).

I do not know if these theories are the dominant ones or not. Just that there are.
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GeorgeH
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### Re: Why does c =c ?

For any given universe there are only two options; either there is a maximum speed limit, or there isn't. We happen to live in the first kind. All of the "weirdness" of special relativity comes directly from that idea, nothing more. This includes the speed of light, which is completely irrelevant to the development of SR.

With that in mind, the question "why doesn't light go faster than the universal speed limit?" doesn't make much sense; a better question might be "why does the speed of light appear to be exactly equal to the universal speed limit?" The answer to that question is we don't know, and further we aren't even completely sure that they are equal - it's possible that light of different energies travels at different speeds only very slightly below the universal speed limit. If they do, though, the difference would be something on the order of a second in the time it takes to travel across pretty much the entire known universe.

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### Re: Why does c =c ?

It makes more sense if you look at it as a wave instead of as a particle.

For example: A sound wave travels at the speed of sound, and only at the speed of sound. It can not accelerate, or decelerate. It has no mass. The speed of sound is determined by various material properties of the transmitting medium, including temperature, molecular mass and whatnot.

Now compare: A light wave travels at the speed of light, and only at the speed of light. It can not accelerate, or decelerate. It has no mass. The speed of light is a constant of nature.
Last edited by You, sir, name? on Fri Apr 02, 2010 6:06 pm UTC, edited 1 time in total.
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Vaniver
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### Re: Why does c =c ?

Taure wrote:That seems a little unsatisfying... I mean, it tells us what occurs, but not why. There must be a reason why the speed of light is the speed it is, and not a different speed.

I guess we just don't know what it is yet.
There are mathematics which describe the world which, among other things, have as a consequence that the speed of light in vacuum is constant, and larger than the speed of light in all other substances.

Why do those mathematics describe the world? That's not my department; take it up with God.
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### Re: Why does c =c ?

I remember reading about a particular (hypothetical) particle that travels faster than light. Now if they actually exist, then C can't be the speed limit. And we should start using tachyon-phones; then people can receive our calls before we call them. Wait, what happens if they receive our call but we decide to not call them? Gah!!
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edgey
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### Re: Why does c =c ?

Light doesn't "accelerate". It always travels at c.
Even in a refractive medium, photons themselves travel at c before being absorbed/reemitted (the act of which slows them down).

I believe that certain properties of vacuum prevent light from traveling faster/slower than it does, I'm only studying Physics at A-level at the moment (final year before undergrad in the UK)

thoughtfully
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### Re: Why does c =c ?

The thing to realize is that c isn't really the speed of light. It got that name for historical reasons. c is a property of spacetime. Photons happen to travel at c due to having no rest mass. Any other such particle would do the same.

The clearest way of seeing c as a maximum speed is to see it as the only speed (in all four dimensions of spacetime):
The norm of the four-velocity is constant, at c. Perfection is achieved, not when there is nothing more to add, but when there is nothing left to take away.
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BlackSails
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### Re: Why does c =c ?

I thought he was asking what does the speed of light have the particular value that is has, rather than being some other value.

The answer is current theory is that we dont know. There are a bunch of parameters that we have no idea why they have the values they have. For instance, you can express the speed of light as: [imath]c=\hbar\alpha/e^2[/imath], where e is the fundamental charge (actually 3 times the more fundamental charge). But that just pushes the question to "why do those constants have the values they have?"

Antimony-120
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### Re: Why does c =c ?

BlackSails wrote:I thought he was asking what does the speed of light have the particular value that is has, rather than being some other value.

The answer is current theory is that we dont know. There are a bunch of parameters that we have no idea why they have the values they have. For instance, you can express the speed of light as: [imath]c=\hbar\alpha/e^2[/imath], where e is the fundamental charge (actually 3 times the more fundamental charge). But that just pushes the question to "why do those constants have the values they have?"

Let's go with

$c = {1 \over{\sqrt{\epsilon_0 \cdot \mu_0}}}$

Where [imath]\epsilon_0[/imath] is the permittivity of free space, and [imath]\mu_0[/imath] is the permeability. What these essentially say is how easy it is to generate an electric or magnetic field in a vacuum. Think of it as the "viscosity" or "rigidity" of the material. A really rigid rod will vibrate much faster than a flexible one. It turns out that w.r.t. electromagnetic waves, space is incredibly rigid.

That is one way of looking at why it's that exact number, but doesn't tell you much about it in relation to special relativity. perhaps a better way of stating it, in light of

Taure wrote:Why does light, which has no mass, only reach the speed it does? What limiting factor prevents it from accelerating faster, given that it has no mass?

Is to state it in special relativity terms:

$E^2 = p^2 c^2 + m_0^2 c^4$

Where E is the energy of a particle, p is the momentum, and [imath]m_0[/imath] is the rest mass. For a particle with no rest mass (there are several besides the photon) we get

$E^2 = p^2 c^2$

Plug in our old freinds kinetic energy and momentum formulas (note that these formulas are NOT using rest mass. m is the relativistic mass)

$E = mv^2, p = mv$

$E^2 = m^2 v^4 = p^2 c^2 = m^2 v^2 c^2$

${E^2 \over p^2} = {m^2 v^4 \over m^2 v^2} = c^2$

$v^2 = c^2 \rightarrow v = c$

Any particle with [imath]m_0 = 0[/imath] travels at v = c. Note that this isn't a maximum speed, this is the ONLY speed it can travel at. It can't go slower. One way to look at it is that c is the absolute limit of speed, and any massless particle will naturally travel at the maximum speed possible. The fact that we call it "the speed of light" instead of "the speed of gravity" or "The speed of gluons" is just because that was the first thing we proved travelled at that speed.

EDIT: I should note that you may be more used to seeing E = (1/2)mv^2, and that technically the above terms are E = [imath]\gamma m v^2[/imath] and [imath]p = \gamma m v[/imath], however those gammas cancel out anyhow, and they go to infinity at v = c, so I'm just skipping a step of cancelling out infinities, which is something that normally one has to consider carefully. But in the interest of not writing an entire book on SR, let's just say it's justified in this case.
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BlackSails
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### Re: Why does c =c ?

Antimony-120 wrote:
Let's go with

$c = {1 \over{\sqrt{\epsilon_0 \cdot \mu_0}}}$

Ok, then why is epsilon the value that it is?

jeppetrost
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### Re: Why does c =c ?

Antimony-120 wrote:$E^2 = p^2 c^2 + m_0^2 c^4$

Where E is the energy of a particle, p is the momentum, and [imath]m_0[/imath] is the rest mass. For a particle with no rest mass (there are several besides the photon) we get

$E^2 = p^2 c^2$

Plug in our old freinds kinetic energy and momentum formulas (note that these formulas are NOT using rest mass. m is the relativistic mass)

$E = mv^2, p = mv$

$E^2 = m^2 v^4 = p^2 c^2 = m^2 v^2 c^2$

${E^2 \over p^2} = {m^2 v^4 \over m^2 v^2} = c^2$

$v^2 = c^2 \Rightarrow v = c$

I'm sorry, but you're wrong, since
$E=\gamma m_0 c^2$
$E^2=p^2 c^2 \Rightarrow m^2 c^4=m^2v^2c^2$
The terms cancel out again, and we're left with $v=c$ for $m_0=0$
But your little trip into "better knowing kinetic energy as" and so on... sorry, but it's wrong.

Edit: Oops, sorry for quoting Blacksails.
Last edited by jeppetrost on Sat Apr 03, 2010 4:46 pm UTC, edited 1 time in total.

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### Re: Why does c =c ?

jeppetrost wrote:I'm sorry, but you're wrong, since
$E=\gamma m_0 c^2$
$E^2=p^2 c^2 \Rightarrow m^2 c^4=m^2v^2c^2$
The terms cancel out again, and we're left with $v=c$ for $m_0=0$
But your little trip into "better knowing kinetic energy as" and so on... sorry, but it's wrong.

I think you missed this part

BlackSails wrote:Plug in our old freinds kinetic energy and momentum formulas (note that these formulas are NOT using rest mass. m is the relativistic mass)
I edit my posts a lot and sometimes the words wrong order words appear in sentences get messed up.

jeppetrost
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### Re: Why does c =c ?

No matter what the mass is, E does certainly not equal m v squared.
Doesn't that old formula $E = mc^2$ say anything? Please say it does..
The kinetic energy (relativistic, mind you) is $E_{kin} = (\gamma-1) m_0 c^2$

Antimony-120
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### Re: Why does c =c ?

He's right, I was going from a hazy memory and not really paying attention to the math (since it's not particularly complex) and my brain skipped a beat there. Really shouldn't try skipping over chuncks that are "easy" and not looking up the formulas. I felt something was wrong, and just figured it was my sweeping a gamma under the rug.

But it's the same result really, and the important part about explaining why that's physically true is still correct.

Anyhow, to continue with the physical descripton (I'll avoid using math since apparently I can't be trusted to pay attention) [imath]\epsilon_0[/imath] is sometimes reffered to as the vacuum dielectric constant. I don't know how much you know about dielectrics, but basically imagine a barbell, one of those ones with a big ball on either end, like you'd see in an old cartoon (I think the bodybuilders in family guy also use them at one point). Now imagine one of those balls is positively charged, and the other negatively charged.

Shove this barbell into an electric field. Let's say the field is pulling the positive left and the negative right. If the barbell is free to swing, it will obviously end up so that the positive ball is left, and the negative ball is right.

Now look at the region between these balls. Ignore the external field for a moment, and lets imagine they're just in space hanging out like that. An electric field will be formed between the two balls, pointing from left to right. But if we go back to the one in the electric field, we notice that the external electric field is right to left. So the barbell generated field is actually partially cancelling the external electric field! (Note: I may have the convention for which way the field points backwards, it's irrelevant the point is that the two fields are opposite directions).

If instead of barbells we have atoms which we can polarize (pull the electrons to the right, nucleus towards the left, but the nucleus is still within the shell of electrons) then we get billions of these little fields. Inside whatever material the atoms are making up the electric field is partially cancelled. This is what we call a dielectric. The more they cancel the field, the stronger the dielectric is, the large we say the dielectric constant is.

So you would THINK that the dielectric constant of a vacuum would be 0. If there's nothing hanging out there to polarize, then obviously there's nothing to cancel any field! And if this were true, then the speed of light would most likely be infinite. We'd have some other issues too, for example

$F_c = {1 \over 4 \pi \epsilon_0}{q_1 q_2 \over r^2}$

which is the force between two charged particles, would also be infinite. Essentially if there isn't SOME resistance to electric fields in a vacuum then the electromagnetic force can ignore distance, since it wouldn't diminish as it travelled through free space (It would still have the inverse square part, as it spread itself over the surface of an expanding sphere, but infinity over an inverse square is still infinity).

So where the hell is this number coming from? What would possibly be "resisting" our electric fields?

Well, it turns out that those virtual particles you may have heard of play a big role here. Virtual particles are these pairs of particles that pop in and out of existance over a very short time. They don't "exist" in quite the way real particles do, but they do have some effects. One of the rules about these particles is that you can only ever produce them in pairs, a particle and it's anti-particle. So for example a proton and anti-proton is an acceptable pair, and an electron and positron (anti electron) is an acceptable pair, but a proton and electron are not. There are reasons for this rule, but for the moment all you need to know is that an anti-particle always has the exact opposite charge of it's particle. You might see where this is going. Pairs of charges, separated by a short distance, appearing in empty space...hey! That's just our barbells again! Sure, there's some strangeness to how and why these pairs appear, and how they're interacting with the real world, but all you need to know is that their brief and odd lives are what gives us a dielectic constant even when there's nothing around to polarize.
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BlackSails
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### Re: Why does c =c ?

I didnt write those posts!

Also, never use relativistic mass.

doogly
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### Re: Why does c =c ?

BlackSails wrote:Also, never use relativistic mass.

Double agreed.
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jeppetrost
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### Re: Why does c =c ?

doogly wrote:
BlackSails wrote:Also, never use relativistic mass.

Double agreed.

Triple agreed.
Why do people insist, that mass increases as velocity does?
I can see, it's easier to just write m instead of [imath]\gamma m_0[/imath] (I did it myself a few posts ago, i think)..
Can't we just say that inertia increases and that this inertia is approximately equal to the mass at low velocities?

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### Re: Why does c =c ?

thoughtfully wrote:The clearest way of seeing c as a maximum speed is to see it as the only speed (in all four dimensions of spacetime)

Holy shit! That one sentence completely revolutionized how I understand the universe! Thank you!

Shivahn
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### Re: Why does c =c ?

I... I... I think I need to lie down now. The constant velocity thing almost makes too much sense. Yet it also blows my mind and makes no sense.

edgey
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### Re: Why does c =c ?

thoughtfully wrote:The clearest way of seeing c as a maximum speed is to see it as the only speed (in all four dimensions of spacetime)

Wow. I love physics.

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### Re: Why does c =c ?

doogly wrote:The speed of light needs to be the same in all reference frames. That's the essential fact that drives all the other funny business.

Perhaps you may think, why can't I go add some more energy to this light over here? Why can't I add as much as I want? You can! But that just raises its frequency. The speed doesn't change.

What would be the maximum frequency? I mean, when would the frequency interfere with itself because 2 periods are so close together that they would overlap.

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### Re: Why does c =c ?

Ok, here's a question. If light has energy, that energy must also be mass, as per E=mc^2. But if a beam of light had mass, it wouldn't travel at the speed of light. But it does. How come?

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### Re: Why does c =c ?

Tsjernobyl wrote:
doogly wrote:The speed of light needs to be the same in all reference frames. That's the essential fact that drives all the other funny business.

Perhaps you may think, why can't I go add some more energy to this light over here? Why can't I add as much as I want? You can! But that just raises its frequency. The speed doesn't change.

What would be the maximum frequency? I mean, when would the frequency interfere with itself because 2 periods are so close together that they would overlap.

Classically? Never.

Non-classically, probably never as well.

jeppetrost
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### Re: Why does c =c ?

Eternal Questionner wrote:Ok, here's a question. If light has energy, that energy must also be mass, as per E=mc^2. But if a beam of light had mass, it wouldn't travel at the speed of light. But it does. How come?

What you have there is your "relativistic mass", and is in fact [imath]\gamma m_0[/imath], and so all you know is, that [imath]E/c^2 = \infty * m_0[/imath] which leaves you some error in your calculator for the mass of the photon. Which is just a terrible result.
This is why you would use the formula [imath]E^2=m_0^2c^4+p^2c^2[/imath] instead.
This formula shows that all particles with rest mass 0 will move at speed c. No relativistic mass. Please.

BlackSails
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### Re: Why does c =c ?

The four velocity magnitude only equals c for timelike observers. For massless particles, the four velocity magnitude is zero!

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### Re: Why does c =c ?

So for the laymen here, from the photos frame of view, it is not moving, you are. Hence from it's point of view it has no relativistic mass, as it is at rest. These forums taught me you can just swap reference frames. It's all physics and stamp collecting.
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Tsjernobyl
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### Re: Why does c =c ?

Tass wrote:
Tsjernobyl wrote:
doogly wrote:The speed of light needs to be the same in all reference frames. That's the essential fact that drives all the other funny business.

Perhaps you may think, why can't I go add some more energy to this light over here? Why can't I add as much as I want? You can! But that just raises its frequency. The speed doesn't change.

What would be the maximum frequency? I mean, when would the frequency interfere with itself because 2 periods are so close together that they would overlap.

Classically? Never.

Non-classically, probably never as well.

Maybe too naive, but what happens with the amplitude of the wave? and should amplitude be interpreted as an amplitude in space? I can imagine a particle sitting on a wave, just following the amplitude and the frequency... since there is no max frequency what happens to it?

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### Re: Why does c =c ?

If the wavelength is on the order of the Planck length, then you'd get messed-up things happening in a non-classical universe.
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### Re: Why does c =c ?

Eternal Questionner wrote:Ok, here's a question. If light has energy, that energy must also be mass, as per E=mc^2. But if a beam of light had mass, it wouldn't travel at the speed of light. But it does. How come?

That logic doesn't quite follow. While you can calculate some sort of equivalent mass for any energy you find, for almost all such energies, the mass you get is going to be sheer nonsense with no connection to reality or physics. This is such a case.

Similarly, while you can calculate the wavelength of anything with a momentum, it is highly questionable whether this is meaningful to apply to something like a truck going 40 mph.
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### Re: Why does c =c ?

scarecrovv wrote:
thoughtfully wrote:The clearest way of seeing c as a maximum
speed is to see it as the only speed (in all four dimensions of
spacetime)

Holy shit! That one sentence completely revolutionized how I understand the universe! Thank you!

This is why I love reading relativity threads: when people "see" the constant 4-velocity thing, and realize it's all just a matter of spacetime geometry.

Technical Ben wrote:So for the laymen here, from the photos frame of view, it is not moving, you are. Hence from it's point of view it has no relativistic mass, as it is at rest. These forums taught me you can just swap reference frames. Generally, you can swap reference frames. However, massless particles don't actually have inertial rest frames: there are no inertial frames in SR in which a photon is at rest.

In fact, this concept was probably more fundamental to the origin of SR than the invariance of light speed was. Einstein knew that there were no experiments that showed an electromagnetic wave as if it were frozen in space, thus he reasoned there was some deep physical principle that prevented such frozen waves from appearing.

On the issue of relativistic mass, I agree that it is confusing (as this thread & others like it amply illustrate ), however, the Usenet Physics FAQ points out that there are some situations where it comes in handy. But it's so easy to make mistakes using it, that it's probably best left to the experts (who generally don't need to use it anyway ).

You, sir, name? wrote:
Eternal Questionner wrote:Ok, here's a question. If light has
energy, that energy must also be mass, as per E=mc^2. But if a beam of
light had mass, it wouldn't travel at the speed of light. But it does.
How come?

That logic doesn't quite follow. While you can
calculate some sort of equivalent mass for any energy you find, for
almost all such energies, the mass you get is going to be sheer
nonsense with no connection to reality or physics. This is such a case.

It's not necessarily nonsense. Consider a large box of low mass floating in space, at rest in the observer's inertial frame. The box has a rest mass of M and contains a lump of matter & a lump of antimatter, each of rest mass m drifting near the centre of the box. So the rest mass of the whole system is M + 2m. When the lump of matter & antimatter meet, they mostly annihilate each other, creating lot of photons & imparting high momenta to the remaining shrapnel, but the rest mass of the whole system will not change from M + 2m until the radiation starts to leave the box.

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