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There is theory that light cone shape does not depend on the reference frame in which it is viewed. So why we draw light cones near black hole differently?

I thought that if I am observing (from the Earth) how light travels from black hole neighbourhood I will always measure speed of light.

Light cones

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There is theory that light cone shape does not depend on the reference frame in which it is viewed. So why we draw light cones near black hole differently?

In general relativity, frames of reference are local, not global. Each of the light cones in your diagram corresponds to a certain local frame of reference. An observer using that frame of reference would draw his/her own light cone as the undistorted one, and would draw the other ones as distorted.

In GR, an observer in one local region of space has no unambiguous way of defining the speed of a distant object. Therefore there is no unambiguous way to say whether the speed of light is wrong, if the ray of light we're talking about is far away from us.

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So this is difference between special relativity and general relativity I didn't know. That's why clocks on GPS satellites work faster than on Earth. –  Pawel Welsberg Jul 21 at 18:25

Light travels along paths with a metric interval of zero. In flat spacetime this would be drawn as a light cone with a 45 degree opening angle in a standard Minkowski space time diagram.Things get a bit weirder in GR when spacetime is curved by mass/energy.

In GR, the concept of an invariant speed of light only applies locally in non-accelerating frames of reference. So locally you see symmetric light cones, but non-locally these can be transformed (though see the comment below) to asymmetric cones with bounding lines that can be bigger or smaller than 45 degrees.

You can probably find some more mathematical approaches to answering this question elsewhere on the site. e.g. Does the speed of light vary in noninertial frames?

But see also

http://www.speed-light.info/speed_of_light_variable.htm

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This isn't quite right. It's true that Einstein originally thought about GR as a generalization of SR to accelerated frames of reference, but that doesn't actually work out well and isn't how relativists today think about it. In flat spacetime this would be drawn as a light cone with a 45 degree opening angle. Even in curved spacetime, it's possible to draw all your light cones with a 45-degree angle. This is what we do in a Penrose diagram. –  Ben Crowell Jul 20 at 22:35
    
True - my oversimplification. I'll edit. –  Rob Jeffries Jul 20 at 23:26

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