Ricci tensor for electromagnetic field

ngkamsengpeter

Electromagnetic fields mostly have a stress-energy tensor in which the trace is zero. Is traceless stress energy tensor always implies Ricci scalar is zero? If yes how to prove that?

elfmotat

Yes. Just contract the EFE's:

[itex]g^{\mu \nu }R_{\mu \nu}-\frac{1}{2}g^{\mu \nu }g_{\mu \nu }R=\kappa g^{\mu \nu } T_{\mu \nu }[/itex]

[itex]R^\mu_{~\mu}-\frac{1}{2}\delta^{\mu}_{~\mu} R=\kappa T^{\mu}_{~\mu}[/itex]

[itex]R=- \kappa T^{\mu}_{~\mu}[/itex]

EDIT: I probably should have said yes, assuming no cosmological constant.

ngkamsengpeter

Quote by elfmotat

Yes. Just contract the EFE's:

[itex]g^{\mu \nu }R_{\mu \nu}-\frac{1}{2}g^{\mu \nu }g_{\mu \nu }R=\kappa g^{\mu \nu } T_{\mu \nu }[/itex]

[itex]R^\mu_{~\mu}-\frac{1}{2}\delta^{\mu}_{~\mu} R=\kappa T^{\mu}_{~\mu}[/itex]

[itex]R=- \kappa T^{\mu}_{~\mu}[/itex]

EDIT: I probably should have said yes, assuming no cosmological constant.

Ok. Thanks. What is the physical meaning of traceless Ricci scalar or stress energy tensor? Why would the electromagnetic field have a traceless stress energy tensor?

torquil

Some info on traceless energy-momentem tensors:

http://www.google.no/url?sa=t&rct=j&...1bJmng&cad=rja

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ngkamsengpeter	Ricci tensor for electromagnetic field Tweet Electromagnetic fields mostly have a stress-energy tensor in which the trace is zero. Is traceless stress energy tensor always implies Ricci scalar is zero? If yes how to prove that?

elfmotat Recognitions: Gold Member	Yes. Just contract the EFE's: [itex]g^{\mu \nu }R_{\mu \nu}-\frac{1}{2}g^{\mu \nu }g_{\mu \nu }R=\kappa g^{\mu \nu } T_{\mu \nu }[/itex] [itex]R^\mu_{~\mu}-\frac{1}{2}\delta^{\mu}_{~\mu} R=\kappa T^{\mu}_{~\mu}[/itex] [itex]R=- \kappa T^{\mu}_{~\mu}[/itex] EDIT: I probably should have said yes, assuming no cosmological constant.

torquil	Ricci tensor for electromagnetic field Some info on traceless energy-momentem tensors: http://www.google.no/url?sa=t&rct=j&...1bJmng&cad=rja