Equivalence principle and light

[vin300]

An accelerating elevator is locally equivalent to a gravitational field. When this is applied to light, it is seen that a horizontal beam of light in the accelerating frame curves and the effect is
same in a gravitational field, but wouldn't this violate the constancy of the speed of light?

 

[Whovian]

No. The light will appear to curve, so it will change direction, but the speed won't change at all.

 

[bahamagreen]

If I understand your question...

When light goes in a straight line what goes at c is the leading edge.
When light curves the direction of the leading edge is tangent to that point on the curve at that moment and additionally subject to the inward deflection at some angle to that tangent ...
So does cuved light slow down its tangent component speed so that both tangent and deflection result in c?

I guess when you measure light in the acellerating elevator or gravitational field, you have to either angle the light "up" so it curves back down to the receiver at the same level (and now takes a longer curved path to measure), or do the measurement "straight" and calculate that the light received is not from the source but from a subsequent point in the curved path (which curving away from the straight line would be longer as well)?

It does seem that if the tangent speed of the light holds at c the deflection of the curve would take the resultant speed of the leading edge >c...? So there must be some math/mechanism that accounts for this to hold it at c for all measures.

 

[PeterDonis]

An accelerating elevator is locally equivalent to a gravitational field. When this is applied to light, it is seen that a horizontal beam of light in the accelerating frame curves and the effect is
same in a gravitational field, but wouldn't this violate the constancy of the speed of light?

The short answer is "no", but the reason why it's no depends on how accurately you want to model things:

(1) In the simplest model, the deflection of the beam of light by gravity is perpendicular to the direction of the light (horizontal light, vertical deflection), so the deflection changes the light's direction but not its speed. This is the type of model that Whovian's post describes. This would work the same for any moving object, not just light; for example, a satellite in a perfectly circular orbit around the Earth moves at a constant speed (but changes direction). So in this simple model, the answer is "no, the speed of light is still constant".

(2) A more complicated model allows for the light not being perfectly horizontal. If the light is moving vertically as well as horizontally, then you have to include the fact that the rate of time flow varies with altitude. (This can also be derived from an "accelerating elevator" type of argument.) That means that the "speed" of the light is now affected by the change in the rate of time flow. In other words, the "constancy of the speed of light" is not really a universal law; it only applies in inertial frames and where gravity is absent. For non-inertial frames and cases where gravity is present, the law has to be generalized; the correct general law is that "light moves on null curves". So in the more complicated model, the answer is "no, the constancy of the speed of light is not violated because it's not the correct law for this case".

 

[PeterDonis]

When light goes in a straight line what goes at c is the leading edge.

Leading edge of what?

So there must be some math/mechanism that accounts for this to hold it at c for all measures.

There isn't--the "speed of light" is *not* always c in the general case. The general law is formulated differently, per my previous post.

 

[bahamagreen]

If light departs at t=0 and you intend to measure the arrival time of that light reflected from a distance, is it not understood that the light departing at t=0 is the "leading edge" of the light, because it must correspond to the "leading edge" of the arriving light which triggers the clock to indicate the elapsed period?

Maybe it's called something else... the leading wave front, the first wave, the photons ahead of all the others...

I could measure the time it takes for a long moving train to go from one station to another by clocking the arrival of its front end at two stations, or its back end, or a distinctive part of it in the train's middle somewhere... but I would not time it based on any mixed random parts of it being "present" at the clock location, because the length of the train would confound the measurement. For example, with a long enough train, doing that I might find the train "present" at both stations at the same time...

 

[PeterDonis]

If light departs at t=0 and you intend to measure the arrival time of that light reflected from a distance, is it not understood that the light departing at t=0 is the "leading edge" of the light, because it must correspond to the "leading edge" of the arriving light which triggers the clock to indicate the elapsed period?

Ah, ok; this is a higher level of detail in modeling the light than I think is necessary for this problem; here it's probably sufficient to consider a "light pulse" whose duration is so short that it can essentially be viewed as a "single photon", a single "object" with a single worldline. What you are describing is more like a "wave train" with an actual length in space; this would have to be modeled not as a single worldline but as a set of worldlines, one for each "point" within the wave train. The "leading edge" would then be the first point. But that's more complication than I think we need here.