Frames can really travel faster than light?

[Buzzworks]

I've read about 'ergosphere' in wiki and surprised at the information presented that ergospheres around very fast spinning black holes can drag space time many times faster than light!

Such that any object that falls into this spinning ergosphere, will also begin to 'accelerate' to match the ergosphere's spin direction (of course, according to the theory, the object is still basically free-falling and only seem to accelerate from an outside observer). Eventually, this object will appear to travel around a spinning black hole at many times faster than light, if it's even possible to see.

I've did further reading about 'space frames' and 'frame dragging'. But my math is not good so I'd like to ask if 'space frames' can move faster than light - a sound and realistic enough theory, thanks!

 

[JesseM]

In general relativity you have to distinguish between measurement of the speed of light in a locally inertial frame of a freefalling observer passing by a light beam (see the equivalence principle), and the speed of light in some global coordinate system which by definition is non-inertial in the curved spacetime of general relativity. Light still always has a speed of c in the first case, and no massive object passing through this local inertial frame can be measured to be moving faster than c, but in the second case light itself may not always have a coordinate speed of c, and massive objects may have coordinate speeds greater than c. Even in special relativity the notion that light travels at c, and that massive objects always move slower than c, only applies in an inertial frame, you can define a non-inertial coordinate system in special relativity where the coordinate speed of light may be different than c.

 

[PAllen]

In general relativity you have to distinguish between measurement of the speed of light in a locally inertial frame of a freefalling observer passing by a light beam (see the equivalence principle), and the speed of light in some global coordinate system which by definition is non-inertial in the curved spacetime of general relativity. Light still always has a speed of c in the first case, and no massive object passing through this local inertial frame can be measured to be moving faster than c, but in the second case light itself may not always have a coordinate speed of c, and massive objects may have coordinate speeds greater than c. Even in special relativity the notion that light travels at c, and that massive objects always move slower than c, only applies in an inertial frame, you can define a non-inertial coordinate system in special relativity where the coordinate speed of light may be different than c.

I have heard of this a few times, never worked through an example. I have a follow up question. In such a coordinate system, it would seem it must still be true that light cones define causality bounds, thus define the physical structure of the spacetime region; and that you would never see an object passing a light signal; and, of course, light doesn't go faster than itself. Looked at this way, 'faster than c' would seem to be an artifact of the coordinate system, and it would make physical sense to rescale (non linearly) the system to make this artifact go away.

Thoughts?

 

[JesseM]

I have heard of this a few times, never worked through an example. I have a follow up question. In such a coordinate system, it would seem it must still be true that light cones define causality bounds, thus define the physical structure of the spacetime region; and that you would never see an object passing a light signal; and, of course, light doesn't go faster than itself. Looked at this way, 'faster than c' would seem to be an artifact of the coordinate system, and it would make physical sense to rescale (non linearly) the system to make this artifact go away.

Thoughts?

It's definitely true that light cones still define causality regardless of your choice of coordinate system. I don't know if it'd be possible to ensure that any coordinate system would measure all light beams to move at c just by "rescaling" the system's definition of distance between points on the same surface of simultaneity though, it could be that the only way to get all light to have a coordinate speed of c would be by actually changing the definition of simultaneity which would result in a totally different coordinate system. And I think talk about space moving "faster than light" happens in the case of coordinate systems that have the nice property that the coordinate velocity of any particle at a point in spacetime can be broken into a simple sum of a local velocity which is always c for light, and a second velocity which can be thought of conceptually as the velocity of "space" at that point--this is true in cosmology where people talk about galaxies moving away from us faster than light, see the third paragraph here. I think that the same is true of the 'waterfall coordinates' discussed here, also known as Gullstrand–Painlevé coordinates in the case of a nonrotating black hole...reading the sections on 'speeds of raindrop' and 'speeds of light' in the wikipedia article, it seems the total speed of either can be broken up into a sort of local speed (1 for light, 0 for the raindrop) minus a factor of $$\sqrt{\frac{2M}{r}}$$ which could be seen as the speed that space is moving at that point. I imagine something similar would be true for the type of "waterfall coordinates" used for a rotating black hole, with the speed of space at each point shown in the animated diagrams on that page discussing waterfall coordinates.

 

[George Jones]

This related to

https://www.physicsforums.com/showthread.php?p=2932702#post2932702.

In special relativity (flat spacetime), there is a standard definition of (spatial) distance that can be applied both locally and globally. In other words, this definition of distance applies to nearby objects, and to objects that are far away. Speed is change in distance divided by elapsed time, so this standard definition of distance can be used to calculated speeds of objects that are near and far. Speeds of objects, near and far, calculated in this way always have the speed of light as their speed limit.

The situation in general relativity (curved spacetime) is far different. Because of spacetime curvature, the definition of (spatial) distance used in the flat spacetime of special relativity can only be applied locally, just as the Earth looks flat only locally. This leads to speeds of nearby objects that limited by the the speed of light, but it say nothing about the behaviour of objects that are far away.

Even though the special relativity definition of distance cannot be applied globally in curved spacetime, there is a standard cosmological definition of distance that is used in the Hubble relationships. Strangely, this cosmological definition of distance can be applied to the flat spacetime of special relativity (Milne universe), and when this is done, it produces a definition of distance (for special relativity) that is different than the standard definition of distance in special relativity!

A different definition of distance gives a different concept of speed, since speed is distance over time. This new concept of speed, even within the context of special relativity, produces speeds of material objects that are greater than the standard speed of light! This definition of the speed of light produces, in both cosmology and in special relativity, speeds that are greater than the standard speed of light.

If v is standard speed in special relativity, and V is cosmological "speed" applied to special relativity, then some corresponding values (as fractions of the numerical value of the standard speed of light) are:

  v                   V
0.200                0.203
0.400                0.424
0.600                0.693
0.800                1.10
0.990                2.65

Even though there can be different definitions of spatial distance, there is no ambiguity with respect to the prediction of experimental measurements. One just has to keep in mind what definition is being used.

 

[PAllen]

Can I always set up a coordinate system and associated metric where coordinate distance is always 'local light seconds'? While it may be inconvenient for many purposes, it would naturally encapsulate the causal structure and never create the impression that light goes faster than itself (light faster than c suggests light faster than itself). It may be necessary that distance is measured differently in different directions, but that is not as bizarre as it might seem. Consider nautical coordinates: positions along the Earth's surface measured in latitude / longitude, depth measured in fathoms. One can write the minkowski metric for such coordinates, everything works fine except for a polar coordinate singularity.

 

[DrGreg]

Can I always set up a coordinate system and associated metric where coordinate distance is always 'local light seconds'? While it may be inconvenient for many purposes, it would naturally encapsulate the causal structure and never create the impression that light goes faster than itself (light faster than c suggests light faster than itself). It may be necessary that distance is measured differently in different directions, but that is not as bizarre as it might seem. Consider nautical coordinates: positions along the Earth's surface measured in latitude / longitude, depth measured in fathoms. One can write the minkowski metric for such coordinates, everything works fine except for a polar coordinate singularity.

Suppose we have a spacetime with a metric

$$ds^2 = A(x)^2 \, c^2 \, dt^2 - B(x)^2 \, dx^2$$​

(Lets keep it simple with just one space dimension.) What you would like to do is define $dT = A(x) \, dt$, $dX = B(x) \, dx$, and get

$$ds^2 = c^2 \, dT^2 - dX^2$$​

But that would be equivalent to saying

$$\frac{\partial T}{\partial t} = A(x)$$
$$\frac{\partial T}{\partial x} = 0$$
$$\frac{\partial X}{\partial t} = 0$$
$$\frac{\partial X}{\partial x} = B(x)$$​

These equations have no solution unless A is constant.

So in this example, where A is not constant, you could get a coordinate system with coordinate distance in local light-seconds but you couldn't have coordinate time in local seconds.

 

[PAllen]

Thanks, DrGreg. That clears it up. In fact, that forced me to realize it should have been obvious that I couldn't do what I wanted:

If a coordinate transform could could make all null geodesics have coordinate speed c, it follows that the metric has been validly transformed everywhere to :

ds**2 = c**2 dt**2 - dx**2 - dy**2 - dz**2

which is possible only in an inertial frame in flat space.