In a discussion concerning:

Physical meaning of non-trivial solutions of vacuum Einstein's field equations

there were a number of answers claiming that the flatness of the Ricci space (Rµv=0) does not necessarily entail the flatness of the Riemannian spacetime. This is explained by the fact that besides the Ricci tensor, the Riemann curvature tensor depends also on the Weyl tensor said to govern the propagation of gravitational radiation. And although Rµv=0, the Weyl tensor can still be non-zero.

That being so, it seems that there are some gravitational forces not included in Einstein's field equations, as the Ricci tensor (Rµv) and the Einstein's tensor (Gµv) are said to be trace-reversed (meaning that when one of them vanishes, the other does too). So, if Rµv=0 does not exclude the presence of gravitation - gravitational radiation can still be found through Weyl tensor - then (since Gµv=0) Einstein's gravitational field equations must be incomplete?

Yet there immediately arises another - and in my opinion, more important - question:

What is the source of this gravitational radiation, since Rµv=0 means the whole universe is void of matter?


1 Answer 1


Eh. It's like saying that Maxwell's equations are incomplete because they allow similar solutions. If you start spacetime out with radiation, that radiation will propogate. Just like if you have a universe containing nothing but a light pulse, that pulse will propogate.

  • $\begingroup$ Which means it's sourceless? $\endgroup$ Commented Mar 28, 2014 at 19:22
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    $\begingroup$ @brightmagus: or I guess you could say that the initial conditions are the source. $\endgroup$ Commented Mar 28, 2014 at 19:23
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    $\begingroup$ Meaning "Let there be gravitation"? $\endgroup$ Commented Mar 28, 2014 at 19:30
  • $\begingroup$ I understand that there is an - unanswered - question of the genesis of matter for instance. But then I believed we do not just assume that everything there is can be simply "given". What, in that case, would be the sense of trying to discover all the history of the universe - from BB for example - till now, like fusion, etc.? $\endgroup$ Commented Mar 28, 2014 at 19:36
  • $\begingroup$ @brightmagus: almost every subfield of physics assumes some sort of initial condition and time-evolves it. For the case of BB cosmology, for instance, we assume the universe in the state it is in today, and we time-evolve it backward. $\endgroup$ Commented Mar 28, 2014 at 19:39

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