I learn about a new model to describe the dynamics of particles undergoing diffusion in general relativity. The evolution of the particle system is described by Vlasov equation without friction. The momentum tensor for matter which undergoing diffusion is not differgence-free (or is not preserved), which makes it inconsistent to couple the Vlasov equation to the Einstein equation. As the compensation of this problem, cosmological scalar field added to the left side of the standard Einstein equation. Then the modification of Einstein's field equation becomes: $R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} + \phi g_{\mu\nu} = \kappa T_{\mu\nu}$.

From the calculation obtained the cosmological scalar field $\phi$ satisfies the homogenous wave equation: $\square \phi = 0$, with $\square = \nabla^{\mu} \nabla_{\mu}$.

What I want to ask is how to determine the exact solution of $\phi$ from the above equation?

  • $\begingroup$ Does the field $\phi$ contribute to the energy-momentum tensor? $\endgroup$ – A.V.S. Jan 7 '18 at 8:55
  • $\begingroup$ As far as I know based on the paper I've read, the homogenous wave equation $\square \phi = 0$ implies that energy-momentum tensor of $\phi$: $T^{\phi}_{\mu\nu} = \nabla_{\mu}\phi \nabla_{\nu}\phi - 1/2 g_{\mu\nu} \nabla^{\alpha}\phi \nabla_{\alpha}\phi$, so that divergence-free satisfied : $\nabla^{\nu}T^{\phi}_{\mu\nu}$. But, this energy-momentum tensor is not added as a source in the modification of Einstein's field theory, due to the interpretation of $\phi$ as a background medium rather than an ordinary matter field. $\endgroup$ – ratnamoedya Jan 7 '18 at 13:50
  • $\begingroup$ The original paper is Calogero, S. (2011). A kinetic theory of diffusion in general relativity with cosmological scalar field. J. of Cosmology and Astroparticle Physics, 2011(11), 016, arXiv:1107.4973, and also subsequent works citing it. $\endgroup$ – A.V.S. Jan 7 '18 at 16:11
  • $\begingroup$ Thank you so much for your help, and the paper -of course. I use this paper as one of my reference, include some citations that mentioned in this paper as well. But the problem is, I need to get the exact solution of $\phi$ which satisfies equation $\square \phi = 0$, so that it can be substitute into the modified of einstein's field equation above. I am sorry to have confused you. $\endgroup$ – ratnamoedya Jan 8 '18 at 1:24

Having looked through the original paper, I think that trying to solve the equation $\Box \phi = 0$ is wrong, somehow. The correct equation for $\phi$ would be $$ \nabla_\mu \phi = 3 \sigma J_\mu, $$ where $3\sigma J^\mu=\nabla_\nu T^{\nu\mu}$. The equation $\Box \phi=0$ is a consequence of the following equation for the energy-momentum tensor: $$ \nabla_\mu\nabla_\nu T^{\mu\nu}=0. $$ That last one has kinetic origins, and obtained by integrating over $p$-space.

Of course, equation $\Box \phi=0$ would be useful if you could formulate Cauchy problem for it, but I believe, in general, you could not without solving the whole Vlasov-Fokker-Plank equation in phase space.

  • $\begingroup$ At first I thought that if I got the solution of $\phi$, then 'surely' the equation $\nabla_{\mu}\phi = 3 \sigma J_{\mu}$ is also fulfilled. Moreover the equation $\square \phi = 0$ is similar to many other equations and the solution have been obtained. But then, by reading your explanation, I think what have you said is a plausible suggestion. Thank you very much... And, about your explanation in the last sentence, even to solve Vlasov-Fokker-Planck equation itself isn't easy. New problem arise on how to determine the distribution function of particles $f$. $\endgroup$ – ratnamoedya Jan 10 '18 at 13:57

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