# Gravitational lensing and the speed of light

1. Dec 31, 2011

### hypernihl

Hello all,

Would like to get some feed-back on some questions I had about special/general relativity and a thought experiment.

In the case of gravitational lensing, an obvious case being the earth is lined up with the sun which is lined up with a distant star. From earth the star is seen because of gravitational lensing.

The path of a photon is increased by the curvature of the sun compared to the path it would take if the sun was not present. Is it possible to say that light has been slowed down by the curvature created by the sun? Can this be interpreted as (length contraction)/(time dilation) = c?

Suppose the sun was displaced in space and it's mass was altered such that the earth received direct light and light that curved around the sun. Two photons left the distant star at the same time and at such angles that one photon traveled in a straight line to earth and the other had to curve around the sun.

The direct light would clearly hit the earth first in time. Would this violate special relativity?

Al so, is the constant speed of light a postulate of special relativity or can it be "proven"?

2. Dec 31, 2011

### ghwellsjr

Einstein's theory of Special Relativity is based on two postulates. The first is the principle of relativity which includes the idea that any inertial observer will measure the round-trip speed of light to be the same constant value. The second is that light propagates (one-way) at that same speed. This postulate cannot be proven.

3. Dec 31, 2011

### hypernihl

I was more interested if the following exist in any cases,

(length contraction)/(time dilation) = c

Clearly this would be some sort of conservation of c?

Of course a postulate cannot be proven, You can't prove Newton's (F=ma) or Schrodinger's (i*hbar*delt*psi=H*psi) equations. I just was not sure whether c was derived or postulated?

4. Dec 31, 2011

### tom.stoer

It is dangerous to compare two different spacetimes, ie. to put the 'curved' path of the photon around a mass into a flat spacetime w/o this mass. Mathematically this does not make sense. The path in both spacetimes is always straight; its (4-dim.) length is always minimal.

Definiton of velocity in GR (curved spacetime) is only reasonable locally. You can't define the velocity of a distant object unambiguously (you can't even define its distance unambiguously).

Locally the speed of photon (regardless which mass is present and therefore how the path is 'curved') is always equal to c. That means whereever you put a measuring device you will always measure c.

There is a well-known effect in GR called Shapiro delay which says that deducing the speed of light closed to large masses (like a star or a galaxy) but far away from the observer seems to indicate a delay, a deceleration. It looks as if the light travels at v<c and as if the equations of GR induce something like a refractive index > 1. This effect must always be taken into consideration for astrophysical calculations.

http://en.wikipedia.org/wiki/Shapiro_delay

5. Dec 31, 2011

### ghwellsjr

How could this be a constant? Lengths get smaller at higher speeds and times get longer at higher speeds.
Not even close.
The value of c is always measured to be the same constant but it takes a round-trip of the light to make that measurement. It is not possible to know if the light takes the same amount of time to traverse from the source to a reflector as it does to traverse back from the reflector to the source. Why don't you read the first several sections of Einstein's 1905 paper introducing Special Relativity? In there you will see that Einstein simply declares that the light takes the same amount of time to go each half of the round-trip and this becomes the basis for synchronizing remote clocks in order to establish the concept of a Frame of Reference.

6. Dec 31, 2011

### hypernihl

Yes it makes a straight line locally, but do all those "local" paths add up to the total distance globally?

I see this, but I always see these arguments for time-travel that utilize the fact that space-time is curved and light can get to one point faster than it can another?

Thanks for pointing this out, I'll check it out.

7. Dec 31, 2011

### tom.stoer

The path of light is a straight line globally (!) a so-called geodesic.

I do not understand why you start with science fiction stuff like time-travel. Everything is well-described by GR w/o any sci-fi ...

8. Dec 31, 2011

### hypernihl

So, two photons starting from the same source, each traveling very different paths in space, arrive at the same time?

How is that that sci-fi?

I guess from the feedback I have gotten so far, the question should be, is there a non-zero difference between two photons traveling in flat space and curved space?

9. Dec 31, 2011

### tom.stoer

Again: you can't compare two paths in two different spacetimes; you can only compare two paths within the same spacetime

A non-zero difference of what? How do you define time? The proper time of photons along their world lines is always exactly zero (that's what is meant be 'light-like'). The difference T' between the arrival of two photons emitted at two times with difference T? In a stationary spacetime these differences will be identical T'=T. If T and T' differ this indicates some non-stationary effect, e.g. expanding spacetime - which has nothing to do with gravitational lensing.

Last edited: Dec 31, 2011
10. Dec 31, 2011

### hypernihl

Why? they left the source at the exact same time. One traveled in a straight line the other took a curved path around the sun? What is varying in the ratio of distance over velocity that says the photon travels at v=c?

Flat-space - curved-space ><=(etc) 0?

11. Dec 31, 2011

### tom.stoer

please have a look at my last post; has been extended

see you soon; I am out for a long-distance race ...

12. Dec 31, 2011

### hypernihl

I have to disagree, but good night sir!

13. Dec 31, 2011

### Powd

Mathematically a curved path would take longer than a straight path but what does the curved path have to do with time dilation? I thought time dilation was a contracting of space.

14. Dec 31, 2011

### pervect

Staff Emeritus
The particular scenario you suggest is non-physical, because you can't have the sun there and not-there. But in principle one could arrange a situation with a flash of light from a single source which would be received at two different times at some specific distant location due to gravitational lensing.

This is not be possible in the flat space-time of special relativity, but is possible in the curved space-time of general relativity.

You might try to interpret this as being due to time dilation and whatnot, but it's not really the best way to think about it. It's much better to think about it as curved space-time. Imagine drawing a space-time diagram ( a plot of the position of the photon vs time) for the two paths of the light flash. If you draw it on a flat piece of paper, the two straight lines representing the wordlines of the light will never intersect. If you draw it on a curved piece of paper, and you get the curve just right, they will intersect.

Special relativity basically says that "the speed of light is constant and equal to C everywhere", and that space-time is flat. In contrast, General relativity says that the speed of light is constant and equal to C when it's measured by local clocks and rulers, and that space-time is not necessarily flat.