How Does Space-Time Curvature Affect Light Near Black Holes?

[Dale]

An accelerometer placed on the ground reads g, not 0. Therefore the ground does not follow a geodesic.

If you have a stationary, non-rotating, spherical mass then the spacetime has the Swartzschild metric. In such a spacetime a worldline which is parallel to the time axis (stationary wrt time) is not a geodesic.

 

[A.T.]

If points on a geodesic cannot accelerate, how can a geodesic change direction?

You could put it that way: The geodesic doesn't change direction. But the direction of the dimensions relative to the geodesic changes, in curved spacetime. If you then assume the curved spacetime to be flat, it apears that the geodesic changes direction.

Analogy: Every great circle on the Earth is a geodesic. And when you put a toy car on a globus, you will see that it follows these geodesics without changing direction (steering). But if draw them on a flat map of the earth, some of them are curved, because the flat map of the world is distorted.

In this applet: http://www.adamtoons.de/physics/gravitation.swf you can see both views:
On the left: The curved spacetime forced to be flat by distorting it, also distorting the geodesic which seems to change direction.
On the right: The curved spacetime undistorted, where the geodesic doesn't change direction.

 

[JesseM]

So you don't care if the surface you're projecting onto is actually spacelike everywhere? If not, then the phrase "projected onto three-dimensional space" is pretty misleading.

You've got it!

And it's only misleading if you confuse "space" with "space-like".

But what you are doing is saying we can call absolutely any slice through spacetime "space", which makes the term fairly meaningless. For example, I could take a standard Minkowski coordinate system, relabel the t coordinate as "x" and relabel the x coordinate as "t", and then take a surface of constant t, and if we call this surface "space" and project geodesics onto it, it will be true here as well that geodesics can only radiate in certain directions from an event, forming a cone in "space". But calling this "space" is obviously pretty misleading, since clearly x is a time dimension!

If you insist on my specifying a coordinate system, I choose the following: a series of spheres of test particles fall together through the event horizon. Each zeroes its clock as it passes, say, 1000GM. They fall radially inwards, so each can be given a latitude and longitude. The radius coordinate of any event inside the event horizon is defined as the (proper) time (on its own clock) of the test particle going through that event.

In this coordinate system, a surface of constant t won't be spacelike outside the horizon! After all, if two successive spheres pass the fixed 1000GM sphere and both set their clocks to zero when they do, then a guy on the first sphere can send a message as he passes the 1000GM sphere which will reach a guy on the second sphere before he passes the 1000GM sphere. So, if you take the hypersurface composed of all events that are assigned a time of 0 in this coordinate system, there will be a timelike separation between some of these events.

erm … I didn't define a t coordinate … I left it to the reader to choose one … yours looks ok to me …

You did define a t coordinate, see the quote above. Anyway, if the "space" you're projecting geodesics onto doesn't have to be spacelike, just a surface of constant t, this means that your statement that "Outside an event horizon, the projection of a light-cone is an expanding sphere" while "inside an event horizon, the name is more sensible, because it is a surface which actually does expand outwards inside a cone!" is then not physical statement at all, but just a statement about certain coordinate systems which have the property that t is timelike outside the horizon and spacelike inside. Do you agree that, if we pick a coordinate system where t is timelike both inside and outside the horizon, then the projection of a light-cone onto a surface of constant t will look like an expanding sphere both inside and outside the horizon? And likewise, if we picked a weird coordinate system where t is timelike inside the horizon but not outside, then the projection of a light cone would look like a cone outside but it would look like an expanding sphere inside?

Also, what do you mean when you say mine "looks ok" to you? If you're referring to the one I described here:

I think you could solve this problem by having a set of ordinary clocks fixed at the 1000GM sphere, and then each successive falling sphere sets its own clocks to match the current readings on these fixed-radius clocks at the moment it passes the fixed 1000GM sphere, instead of zeroing its clocks as it passes the fixed sphere like you suggested.

...then as I said, I think the t coordinate would be timelike both inside and outside the horizon in this coordinate system, and therefore projecting the geodesics of light rays from an event onto a surface of constant t would yield expanding spheres both inside and outside the horizon.

Just to be clear, do you mean an electron will overtake a photon even in a locally inertial coordinate system constructed out of freefalling rulers and clocks using the same procedure as inertial coordinate systems constructed out of inertial rulers and clocks in SR? If you are claiming that, you're badly mistaken--this would amount to a denial of the equivalence principle.

Bingo!

It will hit the singularity first.

I didn't ask which would hit the singularity first. I asked whether you're saying the electron's velocity at any given time, as measured using this system of freefalling rulers and clocks in a local region, would be lower than the photon's velocity at the same time. If you are saying that, this would be a violation of the equivalence principle, so I'm confident this would be wrong. Of course, it might still be true that the electron's spatial velocity as measured in this freefalling locally inertial system was lower than the photon's, yet it hits the singularity first in Schwarzschild coordinates or some other system.

 

[tiny-tim]

… a fairly short post …

Hi JesseM!

Do you agree that, if we pick a coordinate system where t is timelike both inside and outside the horizon …

But t is spacelike inside the horizon … that's obvious from the sign of the coefficient of dt² in the metric.

The other three dimensions … which are space … happen to be two space-like and one time-like … again from the signs of the coefficients of the metric.

You did define a t coordinate, see the quote {below}.

Goodness … that was 7 days ago …

If you insist on my specifying a coordinate system, I choose the following: a series of spheres of test particles fall together through the event horizon. Each zeroes its clock as it passes, say, 1000GM. They fall radially inwards, so each can be given a latitude and longitude. The radius coordinate of any event inside the event horizon is defined as the (proper) time (on its own clock) of the test particle going through that event.

No … I used those clocks only to define an r coordinate … with that r coordinate, one could choose t or t + r or t - r or various other possibilities as the t coordinate, and I left it unspecified!

My "t coordinate" that seems to have impressed you so much gives a t coordinate of 0 to all events at 1000GM !

I didn't ask which would hit the singularity first …

I maintain that any realistic coordinate system will have faster particles hitting the singularity first.

 

[JesseM]

But t is spacelike inside the horizon … that's obvious from the sign of the coefficient of dt² in the metric.

t of what coordinate system? Schwarzschild coordinates? Of course I agree that t is spacelike inside the horizon in Schwarzschild coordinates, I've said this several times before. The point is that this is not a physical fact about black holes, it is just an artifact of the way Schwarzschild coordinates are constructed, much like the coordinate singularity at the event horizon of Schwarzschild coordinates. Physicists can and do construct other coordinate systems where t is timelike inside the horizon as well as outside. Do you disagree? If not, do you agree with my statement that if we pick a surface of constant t in such a coordinate system, then the projection of light geodesics emanating from an event will go in all directions rather than being confined to a cone, regardless of whether we choose an event inside the horizon or outside?

You did define a t coordinate, see the quote {below}.

Goodness … that was 7 days ago …

Please don't roll your eyes at me, it's rude. I've been away on vacation since Thursday (got back Monday night). But in any case, when you made your statement "erm … I didn't define a t coordinate … I left it to the reader to choose one", that was in response to a post of mine where I quoted the post of yours where you had defined a t coordinate (a post you had written only a few hours earlier), and discussed your choice of coordinates in detail.

My "t coordinate" that seems to have impressed you so much gives a t coordinate of 0 to all events at 1000GM !

It didn't "impress me", in fact I criticized it for exactly this reason: every successive sphere gets assigned a time-coordinate of t=0 when it crosses 1000GM, therefore there will be a timelike separation between different events on a surface of constant t. Please reread my response, noting the part in bold:

If you insist on my specifying a coordinate system, I choose the following: a series of spheres of test particles fall together through the event horizon. Each zeroes its clock as it passes, say, 1000GM. They fall radially inwards, so each can be given a latitude and longitude. The radius coordinate of any event inside the event horizon is defined as the (proper) time (on its own clock) of the test particle going through that event.

In this coordinate system, a surface of constant t won't be spacelike outside the horizon! After all, if two successive spheres pass the fixed 1000GM sphere and both set their clocks to zero when they do, then a guy on the first sphere can send a message as he passes the 1000GM sphere which will reach a guy on the second sphere before he passes the 1000GM sphere. So, if you take the hypersurface composed of all events that are assigned a time of 0 in this coordinate system, there will be a timelike separation between some of these events.

I think you could solve this problem by having a set of ordinary clocks fixed at the 1000GM sphere, and then each successive falling sphere sets its own clocks to match the current readings on these fixed-radius clocks at the moment it passes the fixed 1000GM sphere, instead of zeroing its clocks as it passes the fixed sphere like you suggested. In this case I would think a surface of constant t would be spacelike both inside and outside the horizon, though I'm not sure. If it is, though, then I'm sure that if you project the direction of geodesics emanating from an event inside the horizon onto the surface corresponding to the t-coordinate of that event, then the projected geodesics would go in all directions on the surface from that event!

And you never answered my question of what coordinate system you were talking about when you said "yours looks ok to me"--were you indeed talking about the coordinate system I suggested in the last paragraph of the above quote? If so, can you tell me if you agree with my statement "I think the t coordinate would be timelike both inside and outside the horizon in this coordinate system, and therefore projecting the geodesics of light rays from an event onto a surface of constant t would yield expanding spheres both inside and outside the horizon"?

I maintain that any realistic coordinate system will have faster particles hitting the singularity first.

You're obscuring the issue once again. Even if it's true in Schwarzschild coordinates that the electron hits the singularity before the photon (and this actually depends on which side of the light cone we're looking at), I wasn't asking you about Schwarzschild coordinates, I was asking very specifically about a coordinate system constructed of freefalling rulers and clocks inside the horizon. Just because the electron hits the singularity "first" in Schwarzschild coordinates doesn't mean it'll hit it first in a freefalling rulers/clocks coordinate system, which should be obvious since the t-axis is actually spacelike inside the horizon while a t-coordinate based on freefalling physical clocks must of course be timelike inside the horizon as well as outside (the statement that 'the electron hits the singularity at an earlier t-coordinate in Schwarzschild coordinates' might turn out to be equivalent to something like 'the electron is crushed into a singularity further in the -x direction on the x-axis of the freefalling rulers/clocks system', which makes sense if we keep in mind Greg Egan's point that observers inside the black hole see the approach to the singularity as resembling the collapse of a universe that has the shape of a hypercylinder, the end product of which is a line singularity rather than a point singularity).

There is nothing sacred or holy about Schwarzschild coordinates, they are just one of many possible coordinate systems you can use for dealing with a black hole. And in a local coordinate system constructed out of freefalling rulers and clocks, the laws of physics must look identical to those of SR, so the photon must have a higher speed than the electron. If you deny this, then you have confused yourself with your implicit reliance on Schwarzschild coordinates into denying the Equivalence principle, a pretty serious mistake.

 

[tiny-tim]

Please don't roll your eyes at me, it's rude.

No it's not … it's friendly! You could have replied: "Ah … I've been away on vacation since Thursday (got back Monday night) "

… a post of mine where I quoted the post of yours where you had defined a t coordinate (a post you had written only a few hours earlier), and discussed your choice of coordinates in detail.

No … I only defined a latitude and longitude and a radius coordinate (the latter by means of the proper time of some free-falling massive particle):

They fall radially inwards, so each can be given a latitude and longitude. The radius coordinate of any event inside the event horizon is defined as the (proper) time (on its own clock) of the test particle going through that event.

There is nothing sacred or holy about Schwarzschild coordinates …

Blasphemer! ​

For are not Schwarzschild coordinates the only true coordinates in which we, created outside an event horizon, live and move and have our being?

And in a local coordinate system constructed out of freefalling rulers and clocks, the laws of physics must look identical to those of SR …

I'm going to start a new thread for this … at 106 posts, this thread is far too long … despite your original scepticism … ​

Is there any reason you started a new thread for this?

Now go to: https://www.physicsforums.com/showthread.php?p=1693456#post1693456 …