# I On the invariant speed of light being the upper speed limit

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1. Oct 30, 2016

### Joker93

Hello! I have a question that has been bothering me since I first started learning about Special Relativity:
Given only the Minskowskian metric and/OR the spacetime interval, how can one reach the conclusion that the speed of light is invariant for every observer and how can one conclude that it is also the upper speed limit? Or, more generally, how can one conclude that there exists an upper speed limit?
Thanks!

2. Oct 30, 2016

### andrewkirk

It is a speed limit in the sense that a particle with mass, whose velocity is initially less than the speed of light in an inertial frame (strictly - has a timelike velocity), cannot be accelerated to have a velocity that exceeds the speed of light in that frame (has a spacelike velocity). The reason is that, because the object's mass in that frame increases without limit as its velocity in the frame nears c, it takes infinite energy to make its speed reach c.

To my knowledge, there is no prohibition on particles having spacelike velocities for their entire existence. Such hypothetical particles are called tachyons. If we want to rule out their existence we need to introduce a postulate that explicitly does that. The prohibition cannot be deduced from the other postulates.

3. Oct 30, 2016

### Joker93

So, in reality, we study special relativity postulating that everything follows timelike trajectories? If so, the question remains: what part of the math involved in special relativity shows this property? Is it the Minkowskian metric that implies an upper speed limit? Is it the spacetime interval? On a second thought, we obtain the spacetime iterval through the Minkowskian metric, so the strictly timelike trajectories(or lightlike if we are talking about things that travel with c) must be implied by the metric.
But I can't see this though.

4. Oct 30, 2016

### andrewkirk

It is Newton's third law: Force = Mass times acceleration. Since mass approaches infinity, an infinite force is needed to accelerate a massive particle to the speed of light, which in turn will require an infinite amount of energy.

5. Oct 30, 2016

### Joker93

But since all special relativity can be derived from the metric(or so I read), shouldn't what you are saying be explained with the metric?

6. Oct 30, 2016

### strangerep

It's from the derivation of the symmetry group of transformations which preserves the Minkowski metric; -- more specifically, the subgroup corresponding to velocity boosts in a given direction. After crunching through the math, one finds the (now well-known) velocity addition formula: $$v'' ~=~ \frac{v + v'}{1 + (vv'/c^2)} ~,$$ governing the composition of 2 Lorentz boosts where the respective relative speeds are $v, v'$ respectively. Provided both $v$ and $v'$ are $\le c$, the result $v''$ can never exceed $c$.

This is explained in every textbook on special relativity.

7. Oct 30, 2016

### Staff: Mentor

This part at least is easy. Since $ds^2=-c^2 dt^2 + dx^2 + dy^2 + dz^2$ is invariant and when $ds=0$ the equation is a sphere expanding at the speed of light, that means that anything which is a sphere expanding at c in one frame is also a sphere expanding at c in all other frames.

8. Oct 30, 2016

### robphy

Find the eigenvectors of the boost transformations that preserve the Minkowskian metric. (Do the (1+1)-dimensional case for simplicity.)
(Bonus: what are the corresponding eigenvalues?)

9. Oct 30, 2016

### Staff: Mentor

By noting that these are already contained in the Minkowski metric. In other words, by assuming the Minkowski metric, you are assuming the conclusion you want to draw.

A better question would be, what is the minimum you can assume in order to get to the conclusion you describe? AFAIK the principle of relativity plus the assumption that the laws of physics are invariant under space and time translations and ordinary rotations is sufficient to narrow down the possibilities to two: (1) no invariant speed and no upper limit (Newtonian physics, Galilean invariance), or (2) a single invariant speed which is also an upper limit (SR, Lorentz invariance). I don't have links handy to papers where this is shown, I'll see if I can find some references.

10. Oct 31, 2016

### Orodruin

Staff Emeritus
I would here interject that "relativistic mass" is a largely obsolete concept that is prone to misunderstanding. I and many others have spent a considerable amount of time in pulling the concept out of people who come here with these. In modern relativity, "mass" only refers to the invariant mass and all actual uses of relativistic mass for a particle can be exchanged for total energy. In this case, the explanation would be: "The classical mechanics expression for total energy is changed in relativity, in relativity the total energy approaches infinity as v goes to c."

11. Oct 31, 2016

### greswd

The Minskowskian metric is built entirely around the notion that the speed of light is invariant.

As for your second question, try applying the Minskowskian metric to a tachyon's frame of reference.

In this paper, they end up with imaginary transverse (y & z) space. which is entirely incomprehensible.
http://dinamico2.unibg.it/recami/erasmo docs/SomeOld/RevisitingSLTsLNC1982.pdf

12. Oct 31, 2016

### robphy

Theoretically, yes.
But suppose that one is somehow handed the metric (in some form) without the full theory behind it, the question seems to be how to then obtain the other features.

For example, one could conduct an experiment where folks with wristwatches start at a event and, after traveling with various velocities, stop [or somehow mark] when their wristwatches reach (say) 10 seconds. [Suppose muons [with a finite mean lifetime in its rest frame] could be produced with various velocities in the lab frame... we somehow measure the observed lifetime in the lab frame as a function of their velocity in the lab frame.] The locus of those stop-events would trace out a hyperbola on a position-vs-time graph, providing the metric by an experiment. From that, what can one deduce?

13. Oct 31, 2016

### greswd

That the speed asymptotically reaches but never hits c?

I wish you could get your light clock spinning again too.

14. Oct 31, 2016

### robphy

Right... one would deduce that [or at least suspect it]. Some extrapolation/curve fitting on the experimental data would probably suggest a hyperbola analytically.
But then go further and uncover the rest of special relativity from that. (Maybe one should do the same experiments in the lab moving with different velocities, etc...)

If you click on my username to see my profile, you'll see my light clock avatar animated there. For some reason, it doesn't seem to work in the threads.

15. Oct 31, 2016

### Staff: Mentor

Providing a portion of the metric. But not all of it; to do that you would need to do experiments involving null and spacelike intervals as well as timelike intervals. In other words, you would have to, for example, do experiments with light clocks moving at various speeds, do experiments with lightning strikes hitting both ends of trains, etc. All of those experiments together (plus, for example, experiments confirming Maxwell's Equations and their prediction that electromagnetic waves always travel at the same speed in vacuum relative to all observers) would give you the full Minkowski metric, including the properties the OP asked about.

16. Oct 31, 2016

### greswd

Have all of these experiments (or similar experiments) been conducted before? For example the lightning train one.

17. Oct 31, 2016

### robphy

Certainly, one would have to do a variety of experiments to be exhaustive and to check predictions made as the theory is built up ...
However, if I have the unit circle [the unit hyperbola], that gives me the unit timelike vectors....
as well a definition of orthogonality... the tangent to the "circle" is "orthogonal" to the [radial] unit timelike vector.
(This works in Euclidean space and in the Galilean spacetime [a PHY 101 position-vs-time-graph].)
By comparing diagrams from labs on inertially moving ships doing the same experiment, I can deduce that this orthogonality condition is preserved,
as well as uncover invariant directions [associated with the null eigenvectors] in the diagram. (In the Galilean case, the null eigenvector is along the spatial direction.)
By deducing the transformations that preserve this, I can get the Minkowskian signature.
[Of course, we're making the usual assumptions of isotropy and homogeneity, etc...]

18. Oct 31, 2016

### Staff: Mentor

Einstein did that one in his head.

I don't know that the exact experiments I described have been done in the exact form I described them; but we certainly have mountains of experimental evidence for the Minkowski metric.

19. Oct 31, 2016

### Staff: Mentor

Given only this data, how would you distinguish the Galilean from the Minkowskian case?

20. Oct 31, 2016

### robphy

The story I tell is that [figuratively] Galileo did this wristwatch experiment with observers spanning the range of velocities available by the technology of his day... say the upper speed was the speed of a racehorse. From the collected data, Galileo would extrapolate from the small segment on his position-vs-time diagram to a vertical line ranging to all velocities.

With the range of velocities was large enough to approach the speed of light or with high-precision timepieces, Minkowski [figuratively] could repeat the experiment and distinguish his circle from Galileo's circle. In addition, whereas Galileo would have found that the tangent-lines to his unit circle are parallel [implying absolute simultaneity], Minkowski would see that the tangent-lines associated with different unit radii are in fact not parallel... simultaneity is relative. (If one formulated the corresponding transformation laws ["rotations"], one would find that their eigenvectors are different.... a little more analysis would uncover additivity of angles [arc lengths on the unit circles] and the non-additivity of slopes in Special Relativity, as one has in Euclidean geometry.)