Question

2. Show that the Einstein equation \begin{equation} R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu} \quad (39) \end{equation} can be written in this alternative form: \begin{equation} R_{\mu \nu} = \frac{8 \pi G}{c^4} \left( T_{\mu \nu} - \frac{1}{2} T g_{\mu \nu} \right) + \Lambda g_{\mu \nu} \quad (40) \end{equation}

          2. Show that the Einstein equation
\begin{equation} R_{\mu \nu} - \frac{1}{2} g_{\mu \nu} R + \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} T_{\mu \nu} \quad (39) \end{equation} can be written in this alternative form:
\begin{equation} R_{\mu \nu} = \frac{8 \pi G}{c^4} \left( T_{\mu \nu} - \frac{1}{2} T g_{\mu \nu} \right) + \Lambda g_{\mu \nu} \quad (40) \end{equation}

2. Show that the Einstein equation

Rμν - (1)/(2) gμν R + Λ gμν = (8 π G)/(c^4) Tμν (39)
can be written in this alternative form:

Rμν = (8 π G)/(c^4)( Tμν - (1)/(2) T gμν) + Λ gμν (40)

Added by Miguel T.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Pritesh Ranjan

Step 1

Now, let's expand the Einstein tensor Gμν using its definition: Gμν = Rμν - 1/2 gμν R where Rμν is the Ricci tensor and R is the scalar curvature. Substituting this expression for Gμν back into the Einstein equation, we get: Rμν - 1/2 gμν R = 8πG Tμν Now, Show more…

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