Understanding Einstein's Law of Gravitation in Empty Space

In summary, there is often confusion around whether Einstein's law of gravitation for empty space should be identified as Ricci tensor=0 or Einstein tensor=0. However, in vacuum, the Einstein equations are R_{ab} - \frac{1}{2}g_{ab}R = 0, which can be contracted to R=0. In 4+ dimensions, this results in R_{ab}=0, while in 2 dimensions, it only implies the contraction, not the tensor itself being zero. Therefore, the equivalence of these conditions is true in all dimensions except for 2, where the Einstein tensor is always zero.
  • #1
snoopies622
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Why is Einstein's law of gravitation for empty space sometimes identified as Ricci tensor=0 instead of Einstein tensor=0. The first condition implies the second one, but not the other way around.
 
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  • #2
snoopies622 said:
Why is Einstein's law of gravitation for empty space sometimes identified as Ricci tensor=0 instead of Einstein tensor=0. The first condition implies the second one, but not the other way around.

Not true. In vacuum the Einstein equations are

[tex]R_{ab} - \frac{1}{2}g_{ab}R = 0[/tex].

If you contract this equation using [itex]g^{ab}[/itex], you obtain [itex]R=0[/itex]; if you then substitute this back into the Einstein equations you'll find that [itex]R_{ab}=0[/itex] for flat space.
 
  • #3
For some reason I get that the Einstein tensor for the surface of a sphere is zero, while the Ricci tensor is not. This would be a case of the second condition not implying the first. Have I miscalculated?

Perhaps the equivalence of the two conditions is always true in 4-d space but not 2-d space..
 
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  • #4
snoopies622 said:
Perhaps the equivalence of the two conditions is always true in 4-d space but not 2-d space..

yes that's your problem I think. Its only in 4+ dimensions that you can have curvature in free space.
 
  • #5
OK, now I'm getting that the equivalence does hold in any number of dimensions except two, in which case the Einstein tensor is always zero.

Well, what do you know. Thank you both!
 
  • #6
Oops, that's the contraction of the Einstein tensor=0 in two dimensional space, not the tensor itself.
 

1. What is Einstein's Law of Gravitation in Empty Space?

Einstein's Law of Gravitation in Empty Space, also known as the Einstein Field Equations, is a set of equations developed by Albert Einstein in his theory of general relativity. It describes how matter and energy in a given region of space cause the curvature of spacetime, which in turn determines the motion of other objects in that region.

2. How does Einstein's Law of Gravitation differ from Newton's Law of Universal Gravitation?

Einstein's Law of Gravitation takes into account the curvature of spacetime, while Newton's Law of Universal Gravitation assumes that space is flat. This means that Einstein's law can explain phenomena such as the bending of light around massive objects, which cannot be accounted for by Newton's law.

3. Can Einstein's Law of Gravitation be tested?

Yes, Einstein's Law of Gravitation has been tested and confirmed through various experiments and observations. One notable example is the observation of the bending of starlight during a solar eclipse, which provided evidence for the curvature of spacetime predicted by the theory.

4. Is Einstein's Law of Gravitation applicable only in empty space?

No, Einstein's Law of Gravitation applies to all regions of space, whether they are empty or contain matter and energy. However, in the presence of matter and energy, the curvature of spacetime and the resulting gravitational effects are more pronounced.

5. How does Einstein's Law of Gravitation relate to the concept of gravity?

Einstein's Law of Gravitation is a more comprehensive and accurate explanation of gravity compared to Newton's Law of Universal Gravitation. It explains gravity as a result of the curvature of spacetime caused by the presence of matter and energy, rather than a force acting between objects as described by Newton's law.

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