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If one adds a scalar to the Hilbert action without considering any matter fields,
S = \int {d^nx {\sqrt -g} (R - 2 \Lambda)
one gets the Einstein equations as:
R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} + \Lambda g_{\mu \nu} = 0
Now, one can take
T_{\mu \nu} = - \frac{\Lambda}{8 \pi G} g_{\mu \nu}
as an energy of space-time, and get
G_{\mu \nu} = 8 \pi G T_{\mu \nu}
I have read several times that this energy is considered to be the energy density of empty space. Calculations are then made considering contributions of the ground state of quantum fields (bosons and fermions) leading to different values depending on different assumptions for this calculation.
What I do not understand is why \inline \Lambda g_{\mu \nu} is considered to be related to matter fields, since the defined action above did not include them (did it?). Shouldn’t this term be an energy of, let's say, ‘pure’ space-time?
S = \int {d^nx {\sqrt -g} (R - 2 \Lambda)
one gets the Einstein equations as:
R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu} + \Lambda g_{\mu \nu} = 0
Now, one can take
T_{\mu \nu} = - \frac{\Lambda}{8 \pi G} g_{\mu \nu}
as an energy of space-time, and get
G_{\mu \nu} = 8 \pi G T_{\mu \nu}
I have read several times that this energy is considered to be the energy density of empty space. Calculations are then made considering contributions of the ground state of quantum fields (bosons and fermions) leading to different values depending on different assumptions for this calculation.
What I do not understand is why \inline \Lambda g_{\mu \nu} is considered to be related to matter fields, since the defined action above did not include them (did it?). Shouldn’t this term be an energy of, let's say, ‘pure’ space-time?
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