What is the effect of a neutron star's gravity on vertical rays of light?

[penomade]

It is often said that the paths of light bend, close to a neutron star? What happens if the ray of light is vertical to the surface . If we take into account the spacetime of that inertial body, the constancy of velocity of light holds, (or does it?). But shall we (as outside observers) ever be able to witness a slow motion of light?

 

[pervect]

The effect of gravity on a "vertical" light beam , i.e. one normal to the surface, is to change the observed frequency of the light beam (i.e. the frequency a static observer measures, a static observe being one that's "not moving" relative to said surface.

The speed of the light does not change, when measured with LOCAL clocks and LOCAL rulers. It's always 'c'.

One way of describing what happens is to say that the clocks "slow down" deeper in a gravity well. This is a rather common explanation, but a bit confusing, in that it is commonly interpreted as there being some notion of absolute time, which flows faster in some places and slower than others. This isn't quite the way it works - thinking this way will expalin some things, but will confuse other things, as it's "not-quite-right".

A rather better analogy is that space-time is curved, like the surface of the Earth. When you draw your space-time diagram on a curved surface, it's possible to have parallelograms where the opposite sides don't have the same length. And this is exactly the situation here. One side of the parallelogram is the time deep in the gravity well, N>1 seconds. Because of the way space-time diagrams are defined, this time interval would be drawn on the diagram as a line, a line whose length represents the amount of time that pased.

Another side of the parallelogram, parallel to the first, is time further up in the gravity well, 1 second.

The remaining two sides of the parallelogram are lines represented the vertical height, spatial lines, that connect the endpoints of the other two.

So to recap:

1) The speed of light doesn't change as you move vertically in a gravitational field, what happens is that the frequency (as measured by a static observer) changes.

2) This change in frequency of the light is known as "gravitational time dilation" and can be best thought of as being due to the curvature of space-time.

 

[penomade]

Thank you Pervect, (I should be very careful, typing 'c'), perfect and lucid, though, I didn't get the answer as to whether it is possible for us (on earth), as a stationary object relative to the star and outside its immense gravity, to view photons leaving the surface of the star, say perpendicular to the line of our sight (and also again vertical to the surface), slower than c (by our measurement).

 

[pervect]

Well, if you're being strict, there isn't any truly unambiguous definition of "remote velocity". For instance, see Baez's paper http://math.ucr.edu/home/baez/einstein/node2.html, the relevant section of which I've quoted below.

You will see people being less than strict all the time, for instance you'll see reports talking about how fast some distant galaxy is receding in cosmology. It's not a huge problem as long as you carefully read how the author is defining "remote velocity". Assuming they do define it - it's more of an issue if they don't bother and make you guess as to the definition they're using. (There's some semi-standard definitions used to report recession velocities to the lay audience in cosmology, for instance, but it may take some digging to uncover them).

It's really not necessary to get into all that though - the notion of a static observer is well defined, and you can always define the velocity locally relative to a static observer at the same point, avoiding the need to talk about remote velocities at all.

In special relativity, we cannot talk about absolute velocities, but only relative velocities. For example, we cannot sensibly ask if a particle is at rest, only whether it is at rest relative to another. The reason is that in this theory, velocities are described as vectors in 4-dimensional spacetime. Switching to a different inertial coordinate system can change which way these vectors point relative to our coordinate axes, but not whether two of them point the same way.

In general relativity, we cannot even talk about relative velocities, except for two particles at the same point of spacetime -- that is, at the same place at the same instant. The reason is that in general relativity, we take very seriously the notion that a vector is a little arrow sitting at a particular point in spacetime. To compare vectors at different points of spacetime, we must carry one over to the other. The process of carrying a vector along a path without turning or stretching it is called `parallel transport'. When spacetime is curved, the result of parallel transport from one point to another depends on the path taken! In fact, this is the very definition of what it means for spacetime to be curved. Thus it is ambiguous to ask whether two particles have the same velocity vector unless they are at the same point of spacetime.

 

[zonde]

But shall we (as outside observers) ever be able to witness a slow motion of light?

Yes we can witness slow motion of light. You might want to read about this experiment - Shapiro delay