If a massive object were to disappear suddenly what effect would it have on the curvature of spacetime.

How do you propose that it should disappear? That effect might have a big impact on what would happen to the curvature.

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That is a question that is often asked in posing a "speed of gravity" question, such as if the sun disappeared, how long would it take the earth(-moon system) to stop orbiting and go off in a straight line. We have had several threads here about this. Short answer is the speed of gravity, the speed of propagation of changes to space-time caused by changes in the stress-energy, the source of the curvature, is 'c'. However, that 'c' doesn't exactly mean what you think it does -- it is a local speed, not a global one. Space-time itself determines how light propagates, and how its own changes propagate. That amounts to some very non-linear behavior in the general case. When the field is weak enough, you can ignore those non-linearites to good approximation, and it propagates just like EM.

That's good ol' General Relativity for you. Things just just don't mean exactly what you think they ought to mean. Gravity is much more complex that EM in this regard. It "extrapolates" the effects of propagation delay to higher order than EM.

So that is the short answer to what may be the larger question you're asking. However the answer to the specific question you're asking is that is cannot be done. You cannot have mass-energy appear or disappear out or into nowhere. That's an invalid source term for the equations of General Relativity.

Which is sort of frustrating, I know. You want to ask what happens if we plop a mass down in flat space-time, and how "long" (in the full case GR, takes your notions of time and space and warps them so, that asking how long something takes is really an ambiguous question) it would take for the curvature changes to propagate.

So you say, well how about if we has a mass sitting somewhere and we accelerate it off at high speed suddenly? How long will it take for space-time to flatten back out. Well, you can't do that either simply.

In Maxwell, on can simply specify a source charge goes accelerating off. EM is concerend with the electromagnetic energy and momentum, and freely allows mechanical (or other) transfers of energy and momentum. EM conserves energy of itself, but it's like a tank with inlets and outlets. It only says the level in the tank is what you put in less what you took out.

GR however, completes that circuit. Mass-energy-momentum is the source of the field. It cares about all forms of energy. So to specify a mass accelerating off, you have to put the sources of the thrust that makes the mass accelerate in as well. Otherwise you a stress-energy source that is invalid.

So what you can do, is say specify the sun is sitting there, and suddenly splits in two, "converting" some of it's own mass to an energy of thrust to kick two halves of itself off rapidly. That will work. Now, a real process to do that would be hard to come by, but as long as it's kosher for energy-momentum conservation, GR is fine with it.

So say the sun splits in two and the two halves go flying off at near light speed. The result will be that the earth will continue to orbit where the sun would've been for about 8 mins until the information about the change has had to time to reach the earth.

-Richard

I understand that in order to conserve energy the sun could not disappear suddenly. The important thing that I am trying to understand is that if the sun were somehow to disappear than the 'dent' which is present in the spacetime would cease to exist. Just like a sheet of rubber-in its case the driving mechanism is elasticity. My question is that in case of spacetime what is the analog driving mechanism. It seems like the answer is 'stress-energy'. What excatly is it?

Well, imagine the sun sitting there, making its little dent. We hit the trigger and two halves of it go flying off in opposite directions at high speed. The change in the dent would propagate out at 'c' according to an external observer (weak-field limit, of course). The space-time around the earth would flatten out as the two halves, and their little dents sped away.

Stress-energy is the name for the source of the gravitational field, the source term of the Einstein Field Equation. It is a 4D, rank-2 tensor, which you can think of as a 4x4 matrix. The 00 term (indices going from 0 to 3), is the energy density, which over c^2 is the mass equivalence. The 0i, and i0 terms (the rest of the first row, and the first column) are the "energy current" or mass current terms, which is also the momentum density components. The remaining ij terms are the momentum flux terms. This latter part, a 3x3 part, is what is known as the stress or momentum tensor (actually pressure tensor -- pressure in the general case is a tensor, not a simple scalar, actually). Together with the energy "4 current" it is called the stress-energy tensor.

All of these, mass-energy, mass-energy current, and momentum, affect the curvature of space-time. But, in the weak field limit, only the 00 term has much effect, and that reduces to our familiar Newtonian gravity.

About "elasticity". To have waves in a medium, you have something that plays the role of a spring constant, and something that plays the role of inertia, giving you second order wave-like coupling. In EM, the spring-like term is the permittivity, epsilon-0, and the "inertia-like" term is the magnetic permeability, mu-0. Their product determines the speed at which changes and waves propagate through. mu*espilon = 1/c^2.

Epsilon has to do with the electric force constant, and mu has to do with the magnetic force constant. In GR, it is 'G', the gravitational constant that plays the role of epsilon, so 'G' is a measure of the "elasticity" of space-time to stretching or compression by stress-energy. The "gravitomagnetic" constant isn't usually written out explicitly, but it is simply ~ G/c^2 (which is a very small number, which explains why gravitomagnetic effects are so vanishingly small in the familiar terrestrial regimes). One could write EM in the same way, with mu ~ 1/c^2 *1/epsilon or K/c^2, where K is the Coulomb force constant (absorbing the 4pi in K, of course, which is how it is done with G).

Stress-energy is the source, but it is G and c^2 that have to do with the "elastic" properties of space-time in the sense of how changes propagate.

-Richard

Please explain the terms-'mass equivalence', 'momentum density', 'momentum flux','gravitomagnetic constant'.

A wave propagating through spacetime shows that the spacetime fabric has the property of inertia. Does the gravitational constant the completely defines the property of inertia.

This is an internet board, not a textbook, and I'm not up to trying to sit and write one here.

The properties of space-time that determine how the gravitational field propagates are determined by 'G' and 'c'. While it has behavior reminiscent of a classical medium in how "waves propagate", it is not really any such thing. G and c are the fundamental constants that determine that behavior, at least in in General Relativity.

-Richard

Something that may be interesting. One can of think of the stress-energy tensor, as the 4-momentumflux tensor. The regular 3D momentum(stress) tensor, the "pressure" tensor, of a fluid at a point can be seen as a dyad product of the momentum density and the velocity vector at that point. In a matrix representation, the dot product can be seen as a matrix multiplication of a 1 x n matrix and an n x 1 matrix. That is a row vector times a column vector. The result is a 1 x 1 "matrix" or a scalar.

Well, the dyad just reverses that, a n x 1 times a 1 x n, or a column times a row. The result is an n x n matrix. That is sort of a how a rank-2 tensor "works".

At any rate, the ij component of that 3D stress tensor means the flux of the ith component of momentum in the jth direction (at a point). The diagaonals are the regular pressure terms. T_11 is then the flux of the x component of momentum in the x direction, T_22 is the flux of the y component in the y direction, etc. The off diagonal terms are the "cross" terms.

Now, going to a 4D space-time version, we can still say that, the flux of the ith component of momentum in the jth direction. But the time-like component, the 0th index has a little different meaning. The 0th component of momentum is "energy", and the flux in the 0th (time) "direction" is "density".

So T_00 means the flux of the time component of momentum in the time direction. Which means the energy density. T_01 means the flux of the time component of momentum in the x direction (in Cartesian coordinates). That means the flux of energy in the x direction, or the xth component of the "energy current".

Not, T_10, means the flux of the xth component of momentum in the time direction, which means the density of the xth component of momentum. That's different conceptually that the T_01, but by the symmetry of the dyad operation, those two are equal, and in general T_ij = T_ji

-Richard

That symmetry of the stress-energy tensor, T_ij = T_ji is *broken* by Einstein-Cartan when you have spin present. That has consequences. Yep, leave it to me to steer this staight to where my mind is at the time, E-C and torsion.

In E-C, you have two sets of equations. One is the basic EFE, coupling the metric to the stress-energy as normal. The other is a much more complex mess coupling the torsion to the "spin" tensor (it depends on the metric as well, so these two are highly coupled).

Anyway, the stress-energy tensor can exchange "stuff" with the spin tensor, and that allows it to become non-symmetric, T_ij <> T_ji. So, in E-C the metric itself can become non-symmetric, making for differences that wouldn't be in the EFE alone with the metric itself.

-Richard

Thanks Richard! It seems that my concepts of spacetime have become a lot clearer . I still have few problems related to Tensor Calculus but I am struggling hard to get them hooked into my head.

Back again with the topic!

If F(mn) be a tensor describing an observed electromagnetic field then what components of this rank two tensor (or matrix) represent electric feild and and what components represents magnetic feild.

Why do we have a unified framework of electric feild and magnetic feild. It is obviously because they they have very close relationship. But also is it because the electric feild and magnetic feild do not transform as tensors but the electromagnetic field do transform as tensors.

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http://en.wikipedia.org/wiki/Field_strength_tensor

It is anti-symmetric, and traceless, with the diagonal elements being zero in fact. The components of E are in the F_0i, and the B components are in the F_ij.

The field tensor is type of derivative of the 4-potential, a 4-vector, or rank-1 tensor.

-Richard

We don't know whether there is a dent in spacetime. We know only about the dent in space and "in time" (in time in a form of gravitational time dilation) compensating for the dent in space. It is a remark just to keep it rigorous since it does not change much in your question, just shows that it is not question about the spacetime but only about space and time since it the spacetime is flat there is no dent in it to propagate but there is still something to propagate.