# Time dilation derivation of special relativity

In almost all of the derivations using the postulates of special relativity (SR), we use experiments involving light signals. For example, we make a clock using a light signal or measure lengths using light signals, etc. The reason for doing this is never stated. Why do we do this?

Aren't there any other thought experiments that could help us achieve the expressions for time dilation and length contractions without using light signals?

Is it even possible to derive expressions for the Lorentz transformation without using experiments involving light signals in some form?

Please don't start off with space-time metric. I want to know if such a thing can be done using the two postulates given by Einstein.

-
Don't the two postulates imply the invariance of $ds^2$ and hence the metric? Why do you preclude the space-time metric? – Bill N May 25 '15 at 2:06

Einstein's two postulates of special relativity are:

1) The laws of physics are the same in all inertial frames of reference

2) Light propagating through empty space always appears to go at the same velocity, $c$.

The expressions we're trying to derive from these postulates, eg. the Lorentz transformations, or time dilation all have that constant $c$ in them, so we're going to have to somehow use postulate 2 (where else could that number come from?). But all that postulate 2 tells us is about beams of light travelling through empty space, so we're necessarily going to have to think about that to derive the rest of special relativity.

Alternative derivations do exist, but usually using Maxwell's equations or some other electrodynamics (see Wikipedia on this).

It is possible to derive the Lorentz transformations using just the first postulate, but you have a constant (which is $c$) that needs to be empirically verified. However, this requires additional assumptions.

Ultimately I think the reason that SR is usually derived by thinking about light clocks and rulers is that is it the simplest way, and the easiest to understand since it all just follows from those two simple and fairly intuitive ideas.

-
Thank you, the wiki page helped. – Iota Jul 2 '14 at 17:42
I agree that light clocks and rules is the simplest (or at least one of the simplest) ways to introduce SR. I personally didn't understand SR until I came across light clocks, and friends that I have explained SR to didn't understand it until I showed them light clocks. – Joshua Jul 2 '14 at 17:46
Relativity is not intuitive. It is elegant. Mutual time dilation is completely disconnected from everyday experience but has beautiful symmetry. – PyRulez Jul 2 '14 at 18:59
It is possible to derive the Lorentz transformations using just the first postulate: Can the first postulate, without any other physical laws (e.g. without any velocities), really exclude Galilean invariance in favor of Lorentz invariance? Or are we counting Galilean as $c\to\infty$ Lorentz here? – Chris White Jul 3 '14 at 0:59
@ChrisWhite See the comments below (and in) my answer. The short answer is: No, one needs additional assumptions. – Danu Jul 3 '14 at 11:51

Your request seems to be unreasonable to me: On the one hand, you demand that the answer makes use only of Einstein's postulates. These postulates are:

1. The laws of physics look the same from any inertial frame of reference.
2. The speed of light, $c$, is constant, independent of the particular inertial frame of reference that one is in.

As one can see, the constancy of the speed of light is crucial.

On the other hand, you want the answer to refrain from referring to light (signals), as well as the metric which, I assume, also precludes making use of the invariant interval $ds^2=-dt^2+d\vec{x}^2$. It should be noted that the existence of an invariant interval can simply be derived from Einstein's postulates, but once again this argument depends on a thought experiment which involves light signals---as it should since it is the speed of light that is ascribed a special property in SRT.

Demanding one refers neither to light signals nor the invariant interval effectively rules out any explanation that I am aware of. I hope it is now clear to you why arguments in SRT usually do (and should) refer to light signals.

Of course, if one allows additional assumptions or assumes previous knowledge of e.g. electrodynamics, this changes matters. However, such approaches are ruled out by your requirement to rely only on Einstein's postulates.

-
I'm with Danu on this. If you start from the invariance of the proper time then the equation for time dilation drops out really easily. Doing it any other way is just making unnecessary work. – John Rennie Jul 2 '14 at 15:55
Also ultimately postulate 2 is implied by postulate 1. Now so there must be some derivation of the Lorentz factor, that doesn't use light beam experiments and implies only from postulate 1. – Iota Jul 2 '14 at 17:40
@Iota Note that it is not possible to derive the second postulate from the first without additional assumptions (e.g. isotropy). This, too, is pointed out on the wikipedia page linked by James Clough. Any other derivation relying on e.g. Maxwell's equations or electrodynamics is ruled out on the basis of being allowed to use only Einstein's postulates, as per your question. I therefore do not understand your downvote (I assume it's yours), and think you should reconsider, for I believe my answer is correct – Danu Jul 2 '14 at 17:48
The constancy of speed of light in all reference frames does play a crucial role in the derivation of time dilation. But how can this result (time dilation) be directly carried over to any of the other events that does not involve light but is happening in one frame moving to an observer. A very simple example would be to replace the light bulb in a moving train by a machine that shoots tennis balls in all directions and measure the time it takes for one of the balls to hit the point on the train floor directly below. The time will be the same for both observers. – M. Zeng Dec 25 '14 at 4:47
@Timo Sorry, I'm not sure what you're getting at. Do you have a question? – Danu Dec 25 '14 at 9:14

Yes - of course it is possible. You can visualize a massive particle bouncing between two mirrors and arrive at the same result. However, you will need to step out of the 'massive frame' into a stationary frame (that's the whole point of TRANSFORMATION). The only way that information about the 'massive frame' reached the stationary frame is via light. So - light enters into the picture as a 'measurement necessiy' - at which point it's CONSTANCY is what leads to the mass increase ( or whatever physical quantity you are trying to measure).

-

Time dilation is intrinsically linked to the behavior of 'light', or the impedance of the vacuum for fields, more generally. Not only the propagation of free photons is influenced by the speed of light, but also those of virtual photons that bind an electron to a molecule, as well as the gluons that hold its neutrons together, and so on. There are no measures of time which are independent of 'the speed of light'; all time evolution is the evolution of the fundamental fields, which all appear to propagate at the same speed. Measuring the oscillations of a molecule is also measuring the behavior of light, but in a more indirect manner than by using an apparatus that makes the behavior of the individual photons more explicit.

-

In almost all of the derivations using the postulates of SR, we use experiments involving light signals, for example, we make a clock using light signal or measure lengths using light signals etc. [... Why?]

Mainly because that's supposedly completely unambiguous and self-explanatory:
for each recognizable signal event to consider the corresponding first observational indication by anyone who observed it at all; in other words: to consider the signal front in order to define and evaluate geometric relations between participating signal sources and receivers.

Aren't there any other thought experiments that could help us achieve the expressions for time dilation and length contractions without using light signals.

Any other notions which are found in assorted propositions about "space-time", such as "inertial frame", "speed" or "velocity", "homogeneity" and "isotropy" (or how to quantify given experimental regions differing from these conditions), "curvature", and so on, should therefore be defined in terms of "encounters".

Arguably this can include "encounters" of "recognizable material points" with signal fronts (due to "encounters" being in turn observable) a.k.a. "light signals"; and it certainly seems difficult to me to express any "space-time propositions" referring to "encounters between two or more recognizable material points" alone (for instance "A question about a relation between time-like world lines").

-

One can derive the form of the Lorentz transformation from (1) Galileo's relativity principle and assumptions of spacetime (2) flatness, (3) homogeneity and (4) isotropy as well as of (5) continuity of the transformation between inertial frames. You don't need the Einstein postulate of the constancy of the speed of light. The argument runs as follows. If one derives the most general transformation law consistent with (1) through (5) as well as an assumption of absolute time, then Galilean relativity is the unique transformation law possible between frames. If, however, we relax the assumption of absolute time, then we find that a whole family of transformations is consistent with (1) through (5). Each member of the family is a Lorentz group ($O(1,\,3)$) parameterized by a universal constant $c$ that has the inertial frame invariance property. Galilean relativity is the $c\to\infty$ limiting member of the family.

See my answer here and here for more details and references.

This method does not yield a value for $c$; this value must come from experiment. But we know the speed of light is experimentally found to be inertial-frame-invariant, therefore the relativity that applies to our universe must be $O(1,\,3)$ with $c$ equal to the speed of light. We can then think of the Michelson Morley experiment as showing (1) that we live in a universe with a finite $c$ parameter, (2) $c$ is the speed of light and thus (3) light is mediated by a massless particle.

The first person to propose this approach was Vladimir Ignatowski.

In effect, the above approach is the logical converse of what Einstein did. Einstein took an assumption of the constancy of the speed of light further to Galileo's postulate and inferred the Lorentz transformation and thus relative time. The above approach begins with an assumption of possibly relative time together with Galileo's postulate and derives the constancy of light. It shows special relativity to be what we get from Galilean relativity if we relax the assumption of absolute time.

-