B GR Curvature at Light Cone Surface: Smooth, Bent, Blocked?

[bahamagreen]

For example, the curvature due to a mass; does that curvature continue passing from within to outside the mass's light cone? If so, is the mass subject to the external curvature? If not, does the curvature have a discontinuity at the light cone surface?

 

[Orodruin]

A mass does not have a single light cone. Only events have light cones and a mass's world line consists of an infinite number of events. There is no such thing as "the light cone of a mass".

 

[bahamagreen]

Is there such a well defined thing as the light cone of the event coincident with a point on the world line of the mass?

 

[Orodruin]

A point on the world line of the mass is just the mass at a given time. It is a single event and has a light cone.

 

[bahamagreen]

Alright, is this a correctly formulated question?

For the light cone of an event coincident with a point on the world line of a mass,
does the curvature continue from within to outside the event's light cone?
If so, is the mass subject to the external curvature?
If not, does the curvature have a discontinuity at the light cone surface?

If this is still incorrect, let me explain what I'm trying to understand... maybe this will reveal an error in my thinking...

GR shifts gravitation from a force with a propagation >>c to an instantaneous curvature ; so I'm wondering if the curvature extends outside an event's light cone. It seems that it must do so. But if it does, then it seems some influence from that curvature outside the event's light cone would be in effect within the light cone. But if it does not extend outside, what does it look like at the boundary of the light cone? Since gravitation is not limited by distance it would not go to zero before the boundary, but would need to extend through it, but if it does not extend through the boundary it must have some magnitude before and at the boundary that is zero beyond the boundary.

I guess what I'm wondering is how gravity as curvature is restricted to influence within event's light cones; gravity as curvature seems like it would exist and be globally influential without regard to light cone boundaries. How does curvature behave so as to not extend influence through boundaries which seem to be arbitrary depending on the local event?

 

[pervect]

Alright, is this a correctly formulated question?

For the light cone of an event coincident with a point on the world line of a mass,
does the curvature continue from within to outside the event's light cone?
If so, is the mass subject to the external curvature?
If not, does the curvature have a discontinuity at the light cone surface?

If this is still incorrect, let me explain what I'm trying to understand... maybe this will reveal an error in my thinking...

GR shifts gravitation from a force with a propagation >>c to an instantaneous curvature ; so I'm wondering if the curvature extends outside an event's light cone. It seems that it must do so. But if it does, then it seems some influence from that curvature outside the event's light cone would be in effect within the light cone. But if it does not extend outside, what does it look like at the boundary of the light cone? Since gravitation is not limited by distance it would not go to zero before the boundary, but would need to extend through it, but if it does not extend through the boundary it must have some magnitude before and at the boundary that is zero beyond the boundary.

I guess what I'm wondering is how gravity as curvature is restricted to influence within event's light cones; gravity as curvature seems like it would exist and be globally influential without regard to light cone boundaries. How does curvature behave so as to not extend influence through boundaries which seem to be arbitrary depending on the local event?

To oversimplify, GR forbids the creation of "mass" out of nothing, so that there is no way to create a mass and cause the field (whether you use the curvature tensor or some other description of the field) at infinity to change instantaneously This is rather similar to the way that Maxwell's equations forbid the creation of charge (charge is conserved), so that there is no way to create a charge and cause an instantaneous field change outside the light cone.

Consider the E&M case more closely. What prevents E&M effects from traveling faster than light? Maxwell's equations do. They say that a change in the field propagates at the speed of light, c. One cannot argue that E&M signals travel faster than light. E&M signals are due to a change in the E&M field, and Maxwell's equations constrain such changes to travel within the light cone. Maxwell's equations forbid the creation of charge, Einstein's field equations do something similar for "mass".

The description is oversimplified because it's actually not mass, but the stress-energy tensor, that causes the gravitational field. There is a way to word things that is technically accurate, of course, but at this point I feel it would be a distraction. Feel free to ask if you're interested, and it's not impossible that someone else will decide these details are important and provide them even if you don't ask. But try not to loose sight of the main point if you get sidetracked, that E&M is limited to the speed of light by Maxwell's equations, which describe how changes in the field propagate, and that GR does something similar with Einstein's field equations.

 

[PeterDonis]

I guess what I'm wondering is how gravity as curvature is restricted to influence within event's light cones

Perhaps it will help to make this statement more precise. Here is the more precise version, in two parts:

At any event in spacetime, the spacetime curvature at that event is determined entirely by what is in the past light cone of that event.

At any event in spacetime, what is present at that event can only affect the spacetime curvature in the future light cone of that event.

Notice that neither of the above statements implies that curvature somehow stops at the boundary of the light cones of any event. The overall geometry of spacetime is continuous, and doesn't "stop" at light cones. The light cones are causal boundaries--they tell you which events can be causally connected to which other events. Determining spacetime curvature is just one form of causal connection.

 

[bahamagreen]

Sorry for the confusion; this is difficult. EM is a propagation, so not >c. Gravitational waves (changes in curvature?) also share that same constraint. I'm still mixed up on how curvature is causal if it is not propagated (and only propagated when it changes), if this is true. Doesn't it shape itself locally like a function, without regard to causal propagation because it not changing? Doesn't the global curvature (including outside the observable) determine local curvature? Does curvature only extend as far as the observable universe?

I think what I'm asking is whether curvature at an event is the linear superposition of all sources of contributing curvature (which I think is correct), but then how the contributors are limited so that the spacetime curvature at that event is determined entirely by what is in the past light cone of that event. Changes in curvature, yes, restricted to the past cone; but the "static" curvature is already here inside and there outside the past light cone isn't it?

 

[Orodruin]

I think what I'm asking is whether curvature at an event is the linear superposition of all sources of contributing curvature (which I think is correct)

No, this is incorrect. Einstein's field equations are not linear so you cannot just make a superposition of different contributions.

Changes in curvature, yes, restricted to the past cone; but the "static" curvature is already here inside and there outside the past light cone isn't it?

The light cones of spatially separated events may intersect and the curvature at those two events may therefore partially depend on the same history.

 

[Ibix]

EM is a propagation, so not >c. Gravitational waves (changes in curvature?) also share that same constraint.

The EM analogy to a static gravitational field is the electrostatic field around a charge. You could just as well ask whether the electromagnetic field changes at the surface of the light cone. Generally no, but you could imagine a charge moving inertially then undergoing a sudden acceleration at some event. The electromagnetic field inside the future light cone of that event reflects the new motion, but the field outside does not. It hasn't had time to be aware of the change yet. The same is true of a gravitational field - the field inside the future light cone of some change in the source reflects that change while the field outside does not. So there is a change in curvature across the surface of the light cone of some change of state of the source, but not in general across the surface of the light cone of some randomly chosen event.

how the contributors are limited so that the spacetime curvature at that event is determined entirely by what is in the past light cone of that event. Changes in curvature, yes, restricted to the past cone; but the "static" curvature is already here inside and there outside the past light cone isn't it?

The past light cone of any event stretches all the way back to the Big Bang. Only matter that was inside the past light cone of the event can affect the event (through gravity or otherwise), but that includes a truly huge volume, a huge amount of mass, most of which isn't moving very much so generates what you might think of as a "static" curvature. But it would be more accurate to think of it as the joint field of all the matter in the past light cone of the event.

 

[martinbn]

If you think that curvature is something that travels faster than light (from within to the outside of a light-cone), then that's the wrong way to think about curvature. Perhaps it would be best to clarify for yourself what curvature actually is.

 

[PeterDonis]

I'm still mixed up on how curvature is causal if it is not propagated (and only propagated when it changes), if this is true. Doesn't it shape itself locally like a function, without regard to causal propagation because it not changing? Doesn't the global curvature (including outside the observable) determine local curvature? Does curvature only extend as far as the observable universe?

All of these questions can equally well be asked about the electromagnetic field. If you can definitely answer them for the EM field, the answers will be the same for curvature.