Exploring Spacetime Curvature: The Role of Mass and Gravitational Force

[tut_einstein]

1) Is it always a given that the spacetime curvature will be flat in a region in which there is no mass?
2) Therefore is the curvature directly dependent on the mass in a particular region?
3) Also, what exactly is included in the term "mass"?
4) If there are no matter fields to curve space time, would there be no concept of a gravitational force?

Thanks.

 

[atyy]

1) No. In "outer space" (say surrounding the earth) spacetime is curved, even though there is no matter there.
2) No. Another way to think about it is just as a charge can produce an electric field far from where it is, a mass can affect spacetime curvature far from it.
3) Matter is distinguished from gravity by having localizable energy and momentum. In general, it has other properties in addition to energy and momentum with equations relating those properties, and also an equation relating those properties to its energy and momentum.
4) General relativity does have solutions in which spacetime is curved even without matter. They don't apply to our universe, but they reflect the properties of a theory that does describe our universe with high accuracy.

 

[pervect]

1) No, gravitational waves are a counterexample. They are "ripples" in the space-time curvature that propagate through space-time, once started they become independent of what generated them, like light rays become independent of the acclerating charges that generated them.

2) The curvature in a given region has a part that depends on that region , and another part that does not. If you consider the case of the Earth orbiting the sun, the space-time by the Earth is curved by the sun, even though the sun is distant.

3) Mass may be the source of gravity in Newtonian theory, but in GR, the source of gravity is the stress-energy tensor. Mass is a scalar, a single number, the stress-energy tensor is a 4x4 matrix. So it's not as simple as it was in Newtonian theory.

4) I'm not too sure how to answer that one, maybe someone else will give it a shot.

 

[rationalist76]

1) No. In "outer space" (say surrounding the earth) spacetime is curved, even though there is no matter there.
2) No. Another way to think about it is just as a charge can produce an electric field far from where it is, a mass can affect spacetime curvature far from it.
3) Matter is distinguished from gravity by having localizable energy and momentum. In general, it has other properties in addition to energy and momentum with equations relating those properties, and also an equation relating those properties to its energy and momentum.
4) General relativity does have solutions in which spacetime is curved even without matter. They don't apply to our universe, but they reflect the properties of a theory that does describe our universe with high accuracy.

I thought space-time was curved only by matter and energy? In a vacuum space, would space-time be curved then?

 

[Bill_K]

I thought space-time was curved only by matter and energy? In a vacuum space, would space-time be curved then?

No, honest! That's what #2 said, don't you believe him?

There are many, many vacuum solutions of Einstein's equations that are 'curved' despite containing no matter (stress-energy). In some cases this is because the matter has been idealized as a singularity. An example of this is the Schwarzschild solution that describes black holes, which is a vacuum solution. Other vacuum solutions can be understood by supposing that the field was created by a source that has been moved off to infinity. Radiation solutions are like this - you can have a plane wave with no apparent source. Just as you can have an electromagnetic wave with no apparent source.