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If the Schwarzschild metric is suppose to describe the behaviour of a spherical object in flat space, so the Schwarzschild is different from the flat metric because it describes curved space so why then does the Ricci tensor equal zero. Also, if the metric describes a spherical object of mass $M$ in space why should the Energy-Momentum Tensor vanish. If there is mass then there is energy so why must it vanish.

Qmechanic
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MrDi
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  • possible duplicate of Physical meaning of non-trivial solutions of vacuum Einstein's field equations – JamalS Feb 13 '15 at 07:55
  • JamalS, I have already looked at this question and doesn't really answer my question . – MrDi Feb 13 '15 at 07:58
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    Can you clarify what you're asking? What is the problem with a vanishing Ricci tensor (the Riemann and Weyl tensors don't vanish)? The stress-energy tensor doesn't vanish everywhere - it just vanishes everywhere outside the object. Are you puzzled because it doesn't include the gravitational field self-energy? – John Rennie Feb 13 '15 at 08:08
  • John Rennie, my question is; if the metric describes spherical object in space then shouldn't the energy-momentum tensor be non-zero because the object has mass M, so it has energy. Why do both the Ricci and Energy-Momentum tensors vanish if space is both curved and contains an object of mass M. Yes, what about the object's gravitational field self-energy? – MrDi Feb 13 '15 at 08:13
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    @user28952 You are mistaking being 'curved' with non-vanishing Ricci curvature. The former which is your first intuitive notion of curvature is in fact extrinsic curvature, whilst the Ricci curvature is intrinsic. For example, a cylinder is flat in the Ricci sense, but you see it as 'curved' if say, you fold a piece of paper. This corresponds to non-vanishing extrinsic curvature given by the divergence of the normal. – JamalS Feb 13 '15 at 14:43

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This is an answer to the question as qualified in a comment.

The stress energy tensor is a tensor field so it is a function of position in spacetime. In the Schwarzschild coordinates the geometry is time independent so the local value of the stress-energy tensor is just a function of the position in space. Everywhere outside the spherical object it is zero because there is no mass there. Within the object the Ricci tensor is not zero. For the Schwarzschild black hole all the mass is concentrated at the singularity so the Ricci tensor vanishes everywhere except at the singularity (where it is undefined!).

The stress-energy tensor is a well defined local quantity. My own preferred way to understand the stress-energy tensor is to start with the stress-energy tensor for a point particle, because this is simply:

$$ T_{\mu\nu} = \gamma m v_\mu v_\nu $$

at the position of the particle and zero everywhere else. You build up macroscopic objects by (conceptually) adding up the stress-energy tensors of the point particles that make up those objects. Actually, maybe this is more confusing than helpful - if so ignore the last two paragraphs.

As for the self-energy of the gravitational field, well that's the way the Einstein equation is defined i.e. we don't include the self-energy in the stress-energy tensor. In any case the field self-energy is an evasive quantity and couldn't be written in a local invariant form like the stress-energy tensor.

J.G.
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John Rennie
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  • It should be mentioned that if one were to insist upon having a gravitational self-energy stress-energy tensor, it would be constructed out of a perturbation series of the Einstein tensor. – Ryan Unger Feb 13 '15 at 10:57
  • @John Rennie, I'm puzzled by this line from the source you quote "where $v^{\alpha}$ is the velocity vector (which should not be confused with four-velocity)". Can you clarify? Do they mean it is a 3-vector? – m4r35n357 Feb 13 '15 at 19:51
  • @m4r35n357: oops, $v$ is the coordinate velocity, $d/dt$, not the four velocity, $d/d\tau$. – John Rennie Feb 13 '15 at 19:55
  • @m4r35n357: The four-velocity is the derivative wrt proper time. $v^\alpha$ is the derivative wrt coordinate time. – Ryan Unger Feb 13 '15 at 19:55