How is gravity described mathematically in general relativity?

[knightq]

If I move with constant acceleration, say accelerate my car, in mincowski spacetime (flat), will I feel a inertia force? If so, where it come from? If it comes from gravity, does this conflict with the previous presumption(flat spacetime)?

 

[Dale]

If I move with constant acceleration, say accelerate my car, in mincowski spacetime (flat), will I feel a inertia force?

Yes.

If so, where it come from?

From the fact that your worldline is not a geodesic.

 

[bcrowell]

If so, where it come from? If it comes from gravity, does this conflict with the previous presumption(flat spacetime)?

There is more than one verbal way of describing the situation. DaleSpam has given one.

Another description is that in the accelerating frame, there is a gravitational field. Gravitational fields can exist in flat spacetime. The equivalence principle says that you can't tell the difference between an acceleration and a gravitational field. What can't exist in flat spacetime is curvature.

 

[knightq]

I mean what is the origin of the inertia force.
I know " the fact that your worldline is not a geodesic"

There is more than one verbal way of describing the situation. DaleSpam has given one.

Another description is that in the accelerating frame, there is a gravitational field. Gravitational fields can exist in flat spacetime. The equivalence principle says that you can't tell the difference between an acceleration and a gravitational field. What can't exist in flat spacetime is curvature.

Is it real gravity force? Obviously there is no source to produce this force. And according to what I learn ,the curvature=gravity.

 

[PAllen]

I mean what is the origin of the inertia force.
I know " the fact that your worldline is not a geodesic"


Is it real gravity force? Obviously there is no source to produce this force. And according to what I learn ,the curvature=gravity.

Curvature = tidal gravity; that is, a free falling body will still be under gravitational stress; or a ball of free falling dust will change shape.

As to the source of the force you feel in your accelerating car, it is simply the car pushing on you, caused by the car's wheels pushing on the ground, caused by the transport of force from the engine to the wheels, etc.

 

[PAllen]

Put another way, any force you feel is due to something other than uniform gravity. Something pushing on you (e.g. standing on the ground or accelerating in your car) is typically electromagnetic force. If curvature is significant enough, you feel tidal gravitational stress due to non uniform gravity; however, even this case, you can say it is really electromagnetic force that you feel: different parts of you have inertial paths that tend to change proper distance between your parts; this is resisted by electromagnetic force, which you feel as stress or pressure.

 

[jtbell]

If I move with constant acceleration, say accelerate my car, in mincowski spacetime (flat), will I feel a inertia force?

What exactly do you mean by "inertia force"? Is it different from the force exerted by the car seat against your body?

 

[knightq]

What exactly do you mean by "inertia force"? Is it different from the force exerted by the car seat against your body?

I mean the "false" force opposite to the direction of acceleration, such like the centrifugal force, which you must considerate in the stationary frame.

 

[Dale]

I mean the "false" force opposite to the direction of acceleration, such like the centrifugal force, which you must considerate in the stationary frame.

That is a little different than the question in the OP. For this question, the fictitious force comes from the fact that in your coordinate system the lines of constant coordinate position are not geodesics.

 

[bcrowell]

Is it real gravity force?

For this question to mean something, you would have to define "real."

Obviously there is no source to produce this force.

GR allows all kinds of gravitational effects without sources. For example, there could be primordial gravitational waves that never had a source. In a maximum-entropy big bang, essentially all the energy (measured locally) would be in the form of such waves.

Obviously there is no source to produce this force. And according to what I learn ,the curvature=gravity.

You're oversimplifying. You can have curvature without a force and forces without curvature.

 

[pervect]

I mean what is the origin of the inertia force.
I know " the fact that your worldline is not a geodesic"


Is it real gravity force? Obviously there is no source to produce this force. And according to what I learn ,the curvature=gravity.

What you feel on the seat of your pants is mathematically described by the Christoffel symbols.

The curvature of space-time is mathematically described by the Riemann curvature tensor,

The Riemann curvature tensor can be physically interpreted as the tidal force on a non-rotating and non-accelerating observer.

Tidal forces are what you can actually measure easily. Because there is no gravitationally neutral object to serve as a reference, you can't measure the "force" on a unit mass, the way you can on a unit charge,so you can't use the same technique to define a gravitational force that you use to define an electric field, for example. You can measure the force on a unit charge by comparing the acceleration of a charged particle to a neutral one - but you can't do that with gravity because there isn't anything that's gravitationally neutral.


The tidal forces on an accelerating observer are ALMOST the same as the tidal forces on one who is not accelerating. There's a very, very, small difference, which is usually insignificant but occasionally becomes important if the acclerations are REALLY large (usually it only becomes a problem when they're infinite).

Personally, I usually consider tensor quantities the most "real", which gives the Riemann curvature tensor a slight edge. However, the Riemann can be derived mathematically from an ugly set of non-linear equations involving the Christoffel symbols and their first derivatives, the later determines the former. So you could also consider the Riemann and the Christoffel symbols to be different aspects of the same fundamental phenomenon which we call gravity.

The common idea of gravity as something that you feel on the seat of your pants is more closely related to the Christoffel symbols than it is to the Riemann curvature tensor.

Since most people who aren't trained physicists don't consider tensors as being "the most real", and since people also have very widely varied ideas of what they do consider as being "the most real", the question of "reality" tends to quickly dissolve into a philosphical morass that's not very enlightening to anyone.

Hopefully I've clarified at least a tiny bit some of the mathematical machinery in GR, and what it represents physically.