# Special relativity: Which is the limit of validity in the special theory of relativity?

One of the fundamental assumptions in SR is the law of the constancy of the speed of light in vacuo. But Einstein told us that rays of light are propagated curvilinearly in gravitational fields.

The last statement cannot deal with the fundamental assumption previously written, as the curvature of rays of light can only take place when the velocity of propagation of light varies with position. Actually, that means acceleration.

Okey, once I arrived here I thought it was acceleration the limit of validity of SR. But since I know acceleration is possible (I already asked to this forum about it) in SR I got confused about which is the real limit.

Sources: Relativity: The Special and General Theory

Author: Albert Einstein

Chapter: Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference

• Limit of validity of SR? Well, that is first & foremost an assumption of no gravity. – Qmechanic Sep 10 '17 at 10:34

Light always travels at $c$, no matter what. That first sentence

One of the fundamental assumptions in SR is the law of the constancy of the velocity of light in vacuo.

should actually say

One of the fundamental assumptions in SR is the law of the observed constancy of the velocity of light in vacuo and flat spacetime.

Light can, to a distant observer, appear to bend and slow down in a gravitational field and can even appear to come to a complete stop at the event horizon of a black hole, which is all described in general relativity. However, the light always travels, locally, at $c$.

• Light always has local speed of c. The measured coordinate speed can vary due to a variety of conditions. – dmckee Sep 10 '17 at 16:39
• This answer is kind of a muddle. What would it mean for light to "appear" to do these things? – Ben Crowell Jan 21 '18 at 15:22

As noted in this wiki article: https://en.m.wikipedia.org/wiki/Vacuum_solution_%28general_relativity%29?wprov=sfla1 the limit of validity of SR is zero 'stress energy' and zero 'cosmological constant'.

However, it is much more useful to realise that the minkowski space of SR is also the tangent space of GR, so that over short enough distances and times (which can be made precise) minkowski is a good approximation. Thus SR can be though of as the 'small enough region of spacetime' limit also.

• It would be more straightforward to state the first paragraph in terms of curvature. The curvature just happens to be related to the stress-energy and cosmological constant (or dark energy, if you want to include it as part of the stress-energy). – Ben Crowell Jan 21 '18 at 15:23

I want to clarify the other answers slightly. First of all it is critical to distinguish between velocity and speed: velocity is a vector and speed is just a number. Something can have a constant speed while not having a constant velocity.

Special Relativity is usually taken to say that light travels with constant speed in a vacuum (I am not sure whether this formulation is historically correct). So the question is whether this allows light to travel on curves: yes, it does and here's a demonstration of why.

What we can do is construct a series of devices using suitable mirrors which cause a pulse of light to travel along some polygonal path.

First of all it's obvious that I can make such a device: it doesn't break any of the rules about extended objects in SR (it doesn't need to be rigid or anything like that). Secondly I can make a series of these devices with more and more mirrors so the polygon has more and more sides so the path of the light pulse looks more and more like a continuous curve. And finally, all inertial observers will measure that the speed of the pulse is $c$ along any segment of the polygon.

Well, the limit of such devices as the number of sides goes to infinity is light travelling along a smooth curve, with its speed at any point on that curve, as measured by any inertial observer, being $c$. (This business of taking the limit of a finite number of sides to be not only a polygon with an infinite number of sides but a smooth curve is something mathematicians will twitch at but it's just the same as the thing you do to bootstrap calculus and it's fine.)

So insisting that all inertial observers measure the speed of light to be $c$ does not rule out light travelling on curved paths.

To do that we need a stronger assumption: we need to assume that light travels with constant velocity in a vacuum. For reasons that will become apparent we can reformulate this as saying that light travels along straight lines in a vacuum, and its speed along those lines is $c$. This is fine because the integral curves of constant vectors are straight lines.

And this does rule out light travelling on general curves, because general curves are not straight lines.

So if we make this stronger assumption then, if we observe that light does travel on curvilinear paths we know we have reached some kind of regime in which SR is not valid. And, of course, the brilliant trick to resolve this is to say that, indeed, light does travel along straight lines, but that the spacetime through which it travels is curved by gravity so that 'straight lines' become geodesics.

• This is wrong. Light propagates along null geodesics. A geodesic is the definition of straightness. – Ben Crowell Jan 21 '18 at 15:20
• @BenCrowell The point I am making is that a geodesic is more than constant speed. Indeed the last three paragraphs make this very clear. – tfb Jan 23 '18 at 17:07

rays of light are propagated curvilinearly in gravitational fields

Not true. Rays of light propagate along lightlike geodesics. A geodesic is the definition of straightness.

When you have effects such as gravitational lensing, what you're seeing is the effect of noneuclidean geometry, i.e., the curvature of spacetime. For example, in a curved space, two distinct straight lines can intersect more than once. We tend to visualize this intuitively by visualizing the straight lines as curved, but they're actually straight.

The limit of the validity of SR is that spacetime has to be flat. SR fails when spacetime is curved.

• Imagine a body in a state of uniform rectilinear motion with respect to Galileian reference body K. Then, imagine another body of reference K' (which is accelerated relatively to K). The body will execute an accelerated curved motion with respect to K'. This curvature is due to gravitational field. Now replace the body with a ray of light. That's why I wrote a ray of light is propagated curvilinearly in gravitational fields. Einstein also states that SR works if we disregard the influences of gravitational fields. – JD_PM Jan 24 '18 at 18:06

## protected by Qmechanic♦Sep 10 '17 at 13:01

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