# Why C in Vacuum? Time Dilation Formula Explained

• Ragnar
Water in a moving train will cause a time dilation effect that is different then the person on the platform.f

#### Ragnar

Why is the c in the the time dilation formula have the be the speed of light in a vacuum and not the one in the medium through which an observer sees it move?

Say for example we do that thought experiment used to derive the formula, in a much denser medium(in which the speed of light is still the same to all observers but slower like a diamond). In that case shouldn't c be the speed of light within that medium?

because it is the c in a vacuum that must be observed to be the same c by all observers moving at constant velocities.

Say for example we do that thought experiment used to derive the formula, in a much denser medium(in which the speed of light is still the same to all observers but slower like a diamond).

The speed of light in a medium does depend on the velocity of the medium relative to the observer. Fizeau measured this effect for light in water, as far back as the 1850s.

THen wouldn't that mean that time dilation and therefore all relativistic effects only apply in vacuum? i mean since the speed of light is not the same to all observers in non vacuum.

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THen wouldn't that mean that time dilation and therefore all relativistic effects only apply in vacuum? i mean since the speed of light is not the same to all observers in non vacuum.
No, relativistic effects apply everywhere, not just in a vacuum. All relativity requires (or assumes) is that the speed of light in a vacuum is the same for all observers. (Think of that specific speed as being a kind of universal speed limit.) Lower speeds (including that of light itself in some medium) are not the same for all observers. But relativistic effects still apply.

but why do they still apply? is it because of the postulate that says that all the laws of physics are the same in all inertial frames?

No, you use the familiar value of $c$ when calculating relativistic effects, regardless of whether you're in a vacuum or in a medium. In the context of relativistic equations, you shouldn't think of $c$ narrowly as being "the speed of light", but rather as something like "the universal limiting speed." Light happens to travel at that speed when it's in a vacuum, but not when it's in a medium.

Oh ok i get it now. darn I'm stupid.

SOrry i don't understand why Ragnar is wrong.Can u please explain again thoroughly.

Suppose we perform an experiment with light in vacuum and the same experiment for time dilation in air.Wouldn't they give different results?

FOr ex let's take the time dilation experiment in which a person on a railway platform measures the time taken for light inside a train to traverse some distance.

The event's time is also measured by the person in the train.

The people in vacuum would agree to time dilation.
But in air(say) the equation for time dilation wouldn't apply as both would not observe the speed of light to be constant as it wudve not reached the universal speed limit at which all speeds are non-relative

if you perform the thought experiments in relativity in a medium, then the symmetry is broken. Because, if you are moving in a train underwater, you know you are moving because you can feel the water passing you. (if you are in vacuum that would be impossible) If you know physics well enough, you can take the moving water into account and the take the moving water into account and work out the math, you should get the same result.

how so?the condition for time dilation that the speed of light is the same to each observer will be violated.Wont it?

OK think of the value d now this value is the maximum speed of anything in the universe.

Now using the equation e=md^2

Does that make it easier?

Divorce yourself from speed here, light propagates at c in a vacuum, but c just happens to be a limiting factor also, you could just as easily call it d the speed limit of matter/energy in the universe.

We're not relating e=md^2 to light exactly more to the universal speed limit.

$$t' = \gamma \left(t - \frac{v x}{d^{2}} \right)$$

$$\Delta t' = \gamma \left(\Delta t - \frac{v \Delta x}{d^{2}} \right)$$

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if you perform the thought experiments in relativity in a medium, then the symmetry is broken. Because, if you are moving in a train underwater, you know you are moving because you can feel the water passing you. (if you are in vacuum that would be impossible) If you know physics well enough, you can take the moving water into account and the take the moving water into account and work out the math, you should get the same result.
No that is incorrect Tim Lou.
The water is just as much moving as the train is. No symmetry is broken. The only symmetry breaking is when an observer (or an object) accelerates.

@Schrodinger's Dog

I understand that c is a limit,beyond which relativity ceases to exist.But still if we derive the time dilation equation in water,we won't get the same result

@Schrodinger's Dog

I understand that c is a limit,beyond which relativity ceases to exist.But still if we derive the time dilation equation in water,we won't get the same result
No we won't. As Schrodinger's Dog and others (Doc Al, jtbell) have explained in this thread, the fundamental 'speed limit' of the universe has some value. This value happens to be the same speed at which light propagates in a vacuum. The ultimate speed limit of the universe has the same value irrespective of the medium through which we are travelling. For example it is possible for particles to travel through some medium (other than a vacuum) at a speed greater than the speed at which light propagates through the medium and therefore the speed of light in some medium (other than vacuum) cannot be the ultimate speed limit. For more information search for The Cherenkov Effect and see this http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/cherenkov.html" [Broken]

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I understand that c is a limit,beyond which relativity ceases to exist.
The speed of light in a vacuum, c, is the "speed limit" that defines the connection between space and time between moving frames. That connection is contained in the Lorentz transformations and other equations describing relativistic effects, such as "time dilation".

But still if we derive the time dilation equation in water,we won't get the same result
I think I understand what you are talking about. For example, using pulses of light in a vacuum one can construct a "light clock" and derive the time dilation formula. This derivation is easy, since the speed of light in a vacuum is the same for all observers.

But if you build a light clock in water, the pulses would travel at less than c. You could still derive the formula for time dilation, but you cannot use the simplifying assumption that the light pulses have the same speed for all observers. Instead, you'd have to use the relativistic addition of velocity relations to compute the speed of the light pulses in water for different observers. This makes the derivation much harder! Nonetheless, you will end up with the same time dilation formula.

how so?the condition for time dilation that the speed of light is the same to each observer will be violated.Wont it?
That isn't what the postulate says! As already said above, the postulate says the speed of light in a vacuum is the same for all observers.

All you do by adding a medium to the mix is add another unrelated effect that has to be subracted-out to find the real answer (edit - as Doc said two posts above...). Think about it this way: if a clock measures time by the oscillation of cesium atoms, how does the fact that there is air around the cesium atoms affect how fast they oscillate? It doesn't - it just adds a signal delay for someone watching them oscillate.

In a denser medium, the speed of light would be c "to all observers"
inside the medium with c having a smaller value than the c in vacuum.
The relativistic equations for the new medium would only apply to
inertial frames that have the same density as the new medium.

On the other hand, SR is a result of confusion over performing an experiment
inside a stationary frame and watching the result from a moving frame so
that is a different and probably much bigger problem than the one you are looking at.

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Sorry.I think ur wrong.THe speed of light in a denser medium WILL be relative as it has not reached the non-relative speed barrier which is the speed of light in vacuum

Sorry.I think ur wrong.THe speed of light in a denser medium WILL be relative as it has not reached the non-relative speed barrier which is the speed of light in vacuum

You do know of course that the speed at which light propagates in a medium is still the speed of light, in fact light never propagates at less than the speed of light, it might be an idea to look at the FAQ on the General Physics section to see why this is so. In which case your making a faulty assumption. This effect of the medium or refractive index is caused by the effect below, so this has to be accounted for:-

Do Photons Move Slower in a Solid Medium?

Contributed by ZapperZ. Edited and corrected by Gokul43201 and inha

This question appears often because it has been shown that in a normal, dispersive solid such as glass, the speed of light is slower than it is in vacuum. This FAQ will strictly deal with that scenario only and will not address light transport in anomalous medium, atomic vapor, metals, etc., and will only consider light within the visible range.

The process of describing light transport via the quantum mechanical description isn't trivial. The use of photons to explain such process involves the understanding of not just the properties of photons, but also the quantum mechanical properties of the material itself (something one learns in Solid State Physics). So this explanation will attempt to only provide a very general and rough idea of the process.

A common explanation that has been provided is that a photon moving through the material still moves at the speed of c, but when it encounters the atom of the material, it is absorbed by the atom via an atomic transition. After a very slight delay, a photon is then re-emitted. This explanation is incorrect and inconsistent with empirical observations. If this is what actually occurs, then the absorption spectrum will be discrete because atoms have only discrete energy states. Yet, in glass for example, we see almost the whole visible spectrum being transmitted with no discrete disruption in the measured speed. In fact, the index of refraction (which reflects the speed of light through that medium) varies continuously, rather than abruptly, with the frequency of light.

Secondly, if that assertion is true, then the index of refraction would ONLY depend on the type of atom in the material, and nothing else, since the atom is responsible for the absorption of the photon. Again, if this is true, then we see a problem when we apply this to carbon, let's say. The index of refraction of graphite and diamond are different from each other. Yet, both are made up of carbon atoms. In fact, if we look at graphite alone, the index of refraction is different along different crystal directions. Obviously, materials with identical atoms can have different index of refraction. So it points to the evidence that it may have nothing to do with an "atomic transition".

When atoms and molecules form a solid, they start to lose most of their individual identity and form a "collective behavior" with other atoms. It is as the result of this collective behavior that one obtains a metal, insulator, semiconductor, etc. Almost all of the properties of solids that we are familiar with are the results of the collective properties of the solid as a whole, not the properties of the individual atoms. The same applies to how a photon moves through a solid.

A solid has a network of ions and electrons fixed in a "lattice". Think of this as a network of balls connected to each other by springs. Because of this, they have what is known as "collective vibrational modes", often called phonons. These are quanta of lattice vibrations, similar to photons being the quanta of EM radiation. It is these vibrational modes that can absorb a photon. So when a photon encounters a solid, and it can interact with an available phonon mode (i.e. something similar to a resonance condition), this photon can be absorbed by the solid and then converted to heat (it is the energy of these vibrations or phonons that we commonly refer to as heat). The solid is then opaque to this particular photon (i.e. at that frequency). Now, unlike the atomic orbitals, the phonon spectrum can be broad and continuous over a large frequency range. That is why all materials have a "bandwidth" of transmission or absorption. The width here depends on how wide the phonon spectrum is.

On the other hand, if a photon has an energy beyond the phonon spectrum, then while it can still cause a disturbance of the lattice ions, the solid cannot sustain this vibration, because the phonon mode isn't available. This is similar to trying to oscillate something at a different frequency than the resonance frequency. So the lattice does not absorb this photon and it is re-emitted but with a very slight delay. This, naively, is the origin of the apparent slowdown of the light speed in the material. The emitted photon may encounter other lattice ions as it makes its way through the material and this accumulate the delay.

Moral of the story: the properties of a solid that we are familiar with have more to do with the "collective" behavior of a large number of atoms interacting with each other. In most cases, these do not reflect the properties of the individual, isolated atoms.

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I never said that a photon traveled at less than c in a refracting medium!ut just said that the speed of light as observed by two different frames may not necessarily be the same!

I never said that a photon traveled at less than c in a refracting medium!ut just said that the speed of light as observed by two different frames may not necessarily be the same!

And as said the medium only makes it appear that light is traveling lower than c, when in reality it isn't it's propagating the same but the overall effect is that light is slowed by the effect of the refractive index or cumulative the result of absorption and re-emission as explained above.

If you want to take two different mediums then the above explains it, if two frames of reference then the composition equations account for it.

$$v=\frac{v_1+v_2}{1+v_1 v_2}$$

Where light speed is expressed as a fraction of c. .999c say

or

$$w'=\frac{w-v}{1-wv/c^2}.$$

Sorry I didn't quite understand what you were driving at.

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Instead of thinking of c as the "universal speed limit" i think that it would be more effective in this context to think of it as the "universal constant speed". The reason that c appears in the relativity equations is because it's constant for all observers, not because it the maximum speed (though the fact that it's a maximum speed can be derived from the fact that it's a constant).
Since c is constant it can be used as a conservation factor between distance (meters) and time (seconds) so that they can both be expressed in the same units - this is essential for computing the ST interval which is constant for all observers, by using space and time measurements of one observer.

i got it thnxxx

time dilation and clock synchronization

Why is the c in the the time dilation formula have the be the speed of light in a vacuum and not the one in the medium through which an observer sees it move?

Say for example we do that thought experiment used to derive the formula, in a much denser medium(in which the speed of light is still the same to all observers but slower like a diamond). In that case shouldn't c be the speed of light within that medium?
After reading all the answers you have received IMHO I think that there we have a problem of clock syhnchronization. Einstein's speacial relativity works with clocks synchronized using light signals in vacuum because that is the simplest way!

Its not the simplest way,it is just to use the invariance of speed of light according to me.And the magnitude of speed of light in vacuum if taken in a refracting medium(the same value as in vacuum)

would be invariant with repsect to any observer in the refracting medium

c

Its not the simplest way,it is just to use the invariance of speed of light according to me.And the magnitude of speed of light in vacuum if taken in a refracting medium(the same value as in vacuum)

would be invariant with repsect to any observer in the refracting medium
I think your wording is the same as mine!

well might bE!xD

Let's attempt a related thought experiment. Place a piece of glass with index of refraction n=c/v>1 between a light source and a detector and measure the time-of flight of a photon. Would you conclude that the photon had a rest mass? What if you didn't know the rod was in the path?

Let's attempt a related thought experiment. Place a piece of glass with index of refraction n=c/v>1 between a light source and a detector and measure the time-of flight of a photon.

it's not the same photon. in a material, they're absorbed and re-emitted. that's what slows down the wavefront.

Would you conclude that the photon had a rest mass?

i'm one of these turkeys who don't like the application of the term "massless" to photons without some qualification, but whatever you say or think or whatever i say or think, photons do not have rest mass.

in a material, they're absorbed and re-emitted. that's what slows down the wavefront.

okay, i have to retract that (unless this Wikipedia article is wrong):

It is sometimes claimed that light is slowed on its passage through a block of media by being absorbed and re-emitted by the atoms, only traveling at full speed through the vacuum between atoms. This explanation is incorrect and runs into problems if you try to use it to explain the details of refraction beyond the simple slowing of the signal.

Classically, considering electromagnetic radiation to be like a wave, the charges of each atom (primarily the electrons) interfere with the electric and magnetic fields of the radiation, slowing its progress.

The full quantum-mechanical explanation is essentially the same, but has to cope with the discrete particle nature (see Photons in matter): The E-field creates phonons in the media, and the photons mix with the phonons. The resulting mixture, called a polariton, travels with a speed different from light.
...

Light that travels through transparent matter does so at a lower speed than c, the speed of light in a vacuum. For example, photons suffer so many collisions on the way from the core of the sun that radiant energy can take years to reach the surface; however, once in open space, a photon only takes 8.3 minutes to reach Earth. The factor by which the speed is decreased is called the refractive index of the material. In a classical wave picture, the slowing can be explained by the light inducing electric polarization in the matter, the polarized matter radiating new light, and the new light interfering with the original light wave to form a delayed wave. In a particle picture, the slowing can instead be described as a blending of the photon with quantum excitations of the matter (quasi-particles such as phonons and excitons) to form a polariton; this polariton has a nonzero effective mass, which means that it cannot travel at c. Light of different frequencies may travel through matter at different speeds; this is called dispersion. The polariton propagation speed v equals its group velocity, which is the derivative of the energy with respect to momentum.

so, i think the article you want to look up is the one on polariton.

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From Post 32: "It is sometimes claimed that light is slowed on its passage through a block of media by being absorbed and re-emitted by the atoms, only traveling at full speed through the vacuum between atoms. This explanation is incorrect and runs into problems if you try to use it to explain the details of refraction beyond the simple slowing of the signal."

As one of the posters on this forum, I have long argued against the absorption/re-emission explanation. Its gratifying to see some new views on the subject.

In a similar vein, see the following post from PF's own "Physics Forums FAQ" in the General Physics forum:

https://www.physicsforums.com/showpost.php?p=899393&postcount=4 [Broken]

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it's not the same photon. in a material, they're absorbed and re-emitted. that's what slows down the wavefront.

i'm one of these turkeys who don't like the application of the term "massless" to photons without some qualification, but whatever you say or think or whatever i say or think, photons do not have rest mass.

Thank you for the Wikipedia passage; I had not read it. It shows how we fit refraction into our theory de jour without, perhaps, fully understanding it.

You are correct, of course, in the sense that a free photon doesn't have rest mass. When it interacts with matter it is something else, like a "virtual" photon.

The point of this thought experiment is to think about what goes on in the medium, and what it means to travel slower than c. If you don't know what is going on between the source and the detector, how do you know that the photon isn't traveling slower than c and possessing rest mass? If, instead of an n>1 material, what if the photon bounced back and forth between two mirrors before reaching the detector? That would be similar to your first description of absorption and re-emission.