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Is 'c' a constant in General Relativity?

One of the basic postulates of special relativity has been that the velocity of light (denoted by 'c') has the same value in all inertial frames. If this postulate is wrong then the entire Relativistic Mechanics would go wrong.
Now let us have a closer look at the postulate. It says that c is always a constant in inertial frames of reference. Inertial frames are those frames in which things don't look like they are moving on their own. In other words these are un-accelerated frames in which no external force is felt i.e. the motion of all objects could be explained based on the intrinsic properties. 

In Special relativity the definition of such frames is quite simple. Any frame which moves at a uniform velocity w.r.t an inertial frame is also an inertial frame. In all these inertial frames we have the value of c to be a constant. In fact, what we have in Special Relativity are Global Inertial Frames. What we mean by this is that an inertial frame can have its co-ordinate system extended into infinity. For Ex: If you have a un-accelerated spaceship moving in deep space, then it can think of its frame of reference being extended all over the universe! This is mainly because in Special Relativity Space-time is flat everywhere. 

It is quite obvious that we don't live in a universe which has only inertial frames. We deal more with accelerated frames than with inertial ones. So, a real theory of relativity shall include all frames of reference, both inertial as well as accelerated. This is the General Theory of relativity.

The basic postulates of Special Theory are also valid in General Theory. So c is a constant in all inertial frames in General Relativity too. 

What changes in General Relativity is the scope of an inertial frame itself. Instead of global inertial frames we have only local inertial frames in general relativity. Why?

In General Relativity, space-time is curved due to the presence of mass and gravity is nothing but this geometry of space-time caused by the presence of mass. 
In such a situation every un-accelerated object follows the shortest possible path (called the geodesic) in the curved space-time. This is called freefall. For example: An apple falling down from a tree, Earth going round the sun, etc.
Such frames of reference which are in freefall are the only inertial frames available in General Relativity as these are the only frames in which the effects of acceleration is not felt. A ball at rest on a table in an elevator which is in freefall towards earth will continue to be at rest till the elevator hits the earth. But beware, this inertial frame of reference is not global (unlike in special relativity). They are local in nature and should be confined to only that region where the effects of tidal forces are negligible. For example: If the elevator in freefall towards earth is a few thousand kilometers wide and if you have in it two balls at rest separated by a few hundred kilometers then you will find it that the two balls will move towards each other as the elevator continues to fall and this is due to the curvature of space-time towards the center of earth caused by gravity. See below. 
Surely, such a frame in which no intrinsic explanation could be offered for the motion of the objects in it cannot be considered an inertial frame. So we can consider such an elevator to be an inertial frame only for a small region of space where this tidal effects are negligible. So all we have in the presence of gravity are local inertial frames with a limited range for its inertial properties where the range depends on how strong the local space-time is curved. For ex: the same elevator will have a larger local inertial frame while in freefall towards moon than while in freefall towards a blackhole as the the space-time curvature caused by the former is less than the one caused by the latter.

Animation 1: The tidal effects of gravity cause far away objects to move towards each other in a frame in freefall

Thus in general relativity, The velocity of light is a constant when measured in local inertial frames. Even in case of Gravitational Redshift an increase in the wavelength of light is compensated by the decrease in the frequency of the light, so that c=ul remains a constant.

When we are talking about velocity we know that along with the magnitude, it is also the direction that matters. So for velocity not to change, it is necessary that the direction of light also does not change. This is of course true in special relativity as the space-time considered is flat and light travels in a straight line.

In general relativity however, we know that gravity bends light. So isn't the direction of light changed in such a case and hence its velocity too?

While initially it may look true that the direction of light changes in curved space-time, on a deeper investigation we will find that it is actually the space-time itself that is curved and light just follows that curved path. Remember that gravity curves space-time and the shortest (i.e. the straightest possible) distance between 2 points in the presence of gravity is a curved path called the geodesic. 
Hence it is not that the light which gets bent, but the direction itself is curved and so the direction of light remains unaffected and so does its velocity.

- by Gurudev
MADE IN INDIA
gurudevp@vsnl.net
On 26 November 2002

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