# Does time dilation cause the speed of light to be invariant?

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## Main Question or Discussion Point

I'm trying to understand why the speed of light is the same for all observers. I have found different answers on-line. This page claims that it relates to time dilation.

But consider the following thought experiment: two ships flying at 98% c. Ship A is moving toward the sun, and ship B is moving away from the sun. The moment the two ships pass each other they are in the same gravitational field and are flying at the same speed. Time dilation should be the same for both. It would seem that light would be traveling at different speeds for the two ships, but is nevertheless measured to be the same speed. In this situation, it doesn't seem that differences in time dilation provide the explanation.

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Ibix
You need to invoke time dilation, length contraction and the relativity of simultaneity to completely explain the invariance of the speed of light. You measure speed by measuring the distance travelled in a certain time. Each ship sees the other's clocks running slow, their rulers contracted, and notes that their clocks aren't correctly synchronised. These effects conspire so that each one will agree that the other's measurements lead to the same speed of light. But the two ships will give different reasons for why the measurements were what they were.

More typically, you assert the existence of an invariant speed and use that to derive relativistic effects (edit: you might want to look up the light clock or Einstein's train). The above is just running that backwards.

Leave gravity out of this. Newton's theory can't work in a relativistic universe and you need a lot of maths to handle Einstein's theory of gravity.

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I'm trying to understand why the speed of light is the same for all observers.
That was also Einstein's dilemma. In the early 1900's some of the most famous scientists were struggling with Maxwell's theory of electromagnetism trying to understand why it conflicted with the 'standard' of the day, Newtonian mechanics.
The story is outlined here: https://en.wikipedia.org/wiki/Special_relativity#Reference_frames.2C_coordinates_and_the_Lorentz_transformation

In the Lorentz formulas shown there, t' and x' shown are stating how time and distance vary with relative speed! Despite what everybody "knew" was right back then, it turns out space and time were NOT the constants everybody thought, it was the speed of light that Einstein postulated was the REAL constant.

If you and I are in motion with respect to each other, our measures of time, and our views of each others distances are different. In fact, we don't agree on exactly when something happens. Two events in different locations occurring at the 'same time' is not absolute, it depends on the observers reference frame, the relative speeds of the different observers. This is what the first line of IBIX above post means.

Dale
Mentor
I'm trying to understand why the speed of light is the same for all observers.
This is a hard question to answer because it is not really clear what you are asking and what you are willing to take as a given

For example, you could be convinced that there is an invariant speed and be asking why light happens to travel at that invariant speed.

Or you could be convinced that Maxwells equations describe light and be asking how we go from that to the idea that the speed of light is invariant.

Or you could be convinced that Maxwells equations describe light and be asking how we go from that to the idea that the speed of light is invariant.

I'm not familiar with Maxwell's equations. I understand velocity vectors in classical mechanics, which apparently don't apply in relativity. So why does light behave differently, from say, a tennis ball thrown from a moving bicycle?

PeterDonis
Mentor
2019 Award
why does light behave differently, from say, a tennis ball thrown from a moving bicycle?
You're basically asking why the universe obeys the laws of relativity instead of the laws of Newtonian mechanics. The only real answer to that question is "because it does".

If you want something more, something like a reasonable argument based on accepted premises, you need, as Dale said, to tell us what your accepted premises are. So far all you've really told us is what they aren't.

My physics education was in classical mechanics. More recently I have read Relativity: A Very Short Introduction. I understand the relativity of space and time, although I need to learn more about the relativity of simultaneity.

Is it possible that light does in fact travel at different speeds for different observers, but they measure the speed at the same rate, because, for example, maybe their measuring equipment has been slowed down or sped up, compared to the other observer, because of time dilation or other variable?

PeterDonis
Mentor
2019 Award
Is it possible that light does in fact travel at different speeds for different observers, but they measure the speed at the same rate, because, for example, maybe their measuring equipment has been slowed down or sped up, compared to the other observer, because of time dilation or other variable?
What would be the difference? If the "difference" in the speed of light for different observers is not detectable by any physical measurement, in what sense is it a difference?

Dale
Mentor
I'm not familiar with Maxwell's equations. I understand velocity vectors in classical mechanics, which apparently don't apply in relativity. So why does light behave differently, from say, a tennis ball thrown from a moving bicycle?
It is going to be difficult to describe light without Maxwell's equations. A tennis ball follows Newton's laws, and light follows Maxwell's equations. They are very different equations so they behave very differently.

However, it seems like your question may be less about tennis balls and light and more about velocity addition, which follows the same rule for everything:
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/einvel2.html

Dale
Mentor
I understand the relativity of space and time, although I need to learn more about the relativity of simultaneity.
That is good. If you understand length contraction and time dilation, then you almost have the Lorentz transform.
http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/ltrans.html

From the Lorentz transform then you can easily show that c is invariant as follows:
A pulse of light from the origin travels at c in some frame. This is described as a sphere of radius ct as follows:
$c^2 t^2 = x^2 + y^2 + z^2$
Then, use the Lorentz transform equations to change everything into a different frame and simplify. You will get:
$c^2 t'^2 = x'^2 + y'^2 + z'^2$

So it travels at c in both frames.

FactChecker
Gold Member
My physics education was in classical mechanics. More recently I have read Relativity: A Very Short Introduction. I understand the relativity of space and time, although I need to learn more about the relativity of simultaneity.
Yes. The two observers disagree on what events are simultaneous. So they can not agree on how to set widely separated clocks to the same time. That is the root cause of the disagreement on time and distance. It also makes them both measure the speed of light as the same relative to their own motion. That is also why each observer thinks that the other's distance has shrunk without causing a paradox.

There is a story about Einstein realizing the disagreement of simultaneity when he was on a hill and saw two widely separated clock towers in town. That was a breakthrough.

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stevendaryl
Staff Emeritus
Is it possible that light does in fact travel at different speeds for different observers, but they measure the speed at the same rate, because, for example, maybe their measuring equipment has been slowed down or sped up, compared to the other observer, because of time dilation or other variable?
Every inertial observer has his own notion of distance between objects that are at relative to him. Every inertial observer has his own notion of the time between events. Speed is defined by (distance traveled) $\div$ (trip time). So every inertial observer has the same value for this ratio.

The interesting thing about relativity is that the two numbers--$D =$ distance traveled and $T =$ trip time--are not the same for all observers, even though the ratio is, in the case of a light signal.

Chestermiller
Mentor
In my judgment, for whatever it is worth (probably not much because I'm not a relativity guy), all that has been discussed in this thread so far are the effects rather than the causes. In my judgment, the fundamental cause of all these phenomena is the unique geometry of 4D spacetime.

Chet

Mister T
Gold Member
Is it possible that light does in fact travel at different speeds for different observers, but they measure the speed at the same rate, because, for example, maybe their measuring equipment has been slowed down or sped up, compared to the other observer, because of time dilation or other variable?
You're almost there. You just have to accept the fact that there is no special frame of reference in which to judge whether other clocks run slow or are out of sync, or that meter sticks are shortened. These things are true in all reference frames, so that means there is no basis upon which to claim that light really travels at different speeds in different frames. All you have are meter sticks and clocks, and when you use them to measure speeds you find that there is a maximum possible speed. It follows then, that such a speed must be the same in all reference frames.

Dale
Mentor
all that has been discussed in this thread so far are the effects rather than the causes. In my judgment, the fundamental cause of all these phenomena is the unique geometry of 4D spacetime
I agree with your idea expressed here, but I try to avoid the word "cause" in this context. To me, the right term is "imply". If A causes B then not only does A imply B, but also A occurs before B.

The spacetime geometry does imply the invariance of c, but since it doesn't come before or after then I would just say imply rather than cause.

I have the same trouble with this.

If I am travelling at 0.5C towards the sun and measure the speed of light from it, we get told that it still measures C because of our time dialation and length contraction. But the same reasons can't explain why the speed of light from the Earth, which I am moving away from, still measures C.

But the same reasons can't explain why the speed of light from the Earth, which I am moving away from, still measures C.
Whether you move toward or away from a light source, light still measures the same old 'c'. What would change is the frequency of the light you observe; that is, the same color source will appear as a different color.

Depending on just what your post means, you may also be invoking general relativity, in which gravity may further alter appearances. But nevertheless, locally light is always measured at 'c'.

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stevendaryl
Staff Emeritus
I have the same trouble with this.

If I am travelling at 0.5C towards the sun and measure the speed of light from it, we get told that it still measures C because of our time dialation and length contraction. But the same reasons can't explain why the speed of light from the Earth, which I am moving away from, still measures C.
Yes, it certainly can. There are actually three "relativistic" effects that work together:
1. Relativity of simultaneity: Events that are simultaneous in one frame are not simultaneous in a second frame.
2. Length contraction: An object that has a constant length as measured in its own rest frame will have a shorter length when measured from another frame.
3. Time dilation: A clock at rest in one frame will be measured to be running slow by observers in another frame.
The combination of these three effects result in light having the same speed in every reference frame. Of course, the reasoning actually went the other way: Einstein started with the assumption that light has the same speed in every reference frame, and derived those three effects.

I'm not familiar with Maxwell's equations. I understand velocity vectors in classical mechanics, which apparently don't apply in relativity. So why does light behave differently, from say, a tennis ball thrown from a moving bicycle?

When measuring the speed of light, a small error in clock synchronization causes a large error in measurement, because a small time difference is being measured, because it takes just a small amount of time for the light to travel the distance between the clocks.

When measuring the speed of a slow tennis ball, a small error in clock synchronization does not cause a large error in measurement, because a large time difference is being measured, because it takes a large amount of time for the slow tennis ball to travel the distance between the clocks.

Clock synchronization error and clock synchronization difference are not different in this regard. Small errors and small differences become important as speed increases.

Mister T
Gold Member
If I am travelling at 0.5C towards the sun and measure the speed of light from it, we get told that it still measures C because of our time dialation and length contraction.
Yes, but it's best to work through the details and convince yourself that it's true, rather than accept it on authority.

But the same reasons can't explain why the speed of light from the Earth, which I am moving away from, still measures C.
Why not? The same reasoning does apply. To an observer on Earth your clocks are running slow and your meter sticks are contracted, so you measure the same value for the speed of a light beam as he does, whether the beam runs towards or away from Earth. Or whether you move towards or away from Earth.

Ibix
I have the same trouble with this.

If I am travelling at 0.5C towards the sun and measure the speed of light from it, we get told that it still measures C because of our time dialation and length contraction. But the same reasons can't explain why the speed of light from the Earth, which I am moving away from, still measures C.
Why don't we do the maths? In some frame S we see a light pulse moving along in the x direction. Its position is $(x,t)=( ct,t)$. Now we use the Lorentz transform to determine the location of the light pulse measured in a frame S', moving at speed v in the x direction. That is $$\begin{eqnarray} x'&=&\gamma (x-vt) = \gamma (ct-vt)=\gamma (c-v)t \\ t'&=&\gamma \left(t-\frac { vx}{c^2}\right)=\gamma \left(t-\frac { vct}{c^2}\right)=\gamma \left(1-\frac { v}{c}\right)t \end {eqnarray}$$Substituting the last expression for t' into the last expression for x' we get that $x'=ct'$ - in other words the speed is always c, independent of v. You can make v negative, so this is true whichever way you are travelling with respect to the light pulse (feel free to try $x=-ct$ if that makes you feel better).

Since the Lorentz transforms are the mathematical statement of length contraction, time dilation and the relativity of simultaneity, then we can say that yes, they can explain the constancy of the speed of light.

Why not? The same reasoning does apply. To an observer on Earth your clocks are running slow and your meter sticks are contracted, so you measure the same value for the speed of a light beam as he does, whether the beam runs towards or away from Earth. Or whether you move towards or away from Earth.

Because when your measuring apparatus moves away from light at speed 10 m/s, some effects must cause an "error" of 10 m/s in the result of the measurement.

But when your measuring apparatus moves towards the light at speed 10 m/s, some effects must cause an "error" of -10 m/s in the result of the measurement.

Length contraction and time dilation are quite useless in causing the right "errors", because they:

2: are too small at slow speeds
3: cancel out each other

Mister T
Gold Member
Because when your measuring apparatus moves away from light at speed 10 m/s, some effects must cause an "error" of 10 m/s in the result of the measurement.
You spoke of time dilation and length contraction as the basis for always measuring the same value for the speed of a light beam. That's a calculation involving the division of a distance by a time.

You spoke of time dilation and length contraction as the basis for always measuring the same value for the speed of a light beam. That's a calculation involving the division of a distance by a time.

When we calculate how much time dilation and length contraction change the result of measurement of the speed of a light beam, we add or subtract some speeds... And the effect is zero.

We have been ignoring the effect of relativity of simultaneity on speed measurements, so let's consider it now:

speed measurement result without relativity of simultaneity - speed measurement result with relativity of simultaneity = the effect of relativity of simultaneity on speed measurement

the effect of relativity of simultaneity on speed measurement = -10 m/s when clocks are moving at speed 10 m/s towards the light beam

the effect of relativity of simultaneity on speed measurement = 10 m/s when clocks are moving at speed 10 m/s away from the light beam

Time dilation and length contraction can not be the basis for always measuring the same value for the speed of a light beam, because:
1: they don't care about direction
2: they are too small at slow speeds
3: they cancel out each other
4: relativity of clock synchronization takes care of measuring devices always measuring the same value for the speed of a light beam