# Explaining Einstein's Choice of R_ab - 1/2 g_ab R

• Phymath
In summary, Einstein chose the equation R_ab - 1/2 g_ab R = k T_ab because it expresses how the mass/energy causes curvature and must be a conserved quantity. This is the only second rank tensor that gives an automatic concept of conservation and energy, as shown by the equation T_{uv} = 8 \pi G_{uv}, where \nabla^u T_{uv} is equal to zero. While there are more details and nuances involved, this equation can be found in MTW gravitation.
Phymath
Why did Einstein chose R_ab - 1/2 g_ab R = k T_ab the Right side is basically saying the mass/energy causes the curvature and must be a conserved quantity, but why is the curvature expressed like it is on the LHS i know it is conserved through differentiation but why that and not any other variant of Riemann curvature?

The short answer is that this is the only second rank tensor that gives us an automatic concept of the conservation and energy.

This happens because $T_{uv} = 8 \pi G_{uv}$, so that $\nabla^u T_{uv}$ is equal to zero, since $\nabla^u G_{uv}$=0.

A longer answer involves many subtle points, unfortunately.

found it in MTW gravitation thank you thou

## What is Einstein's choice of R_ab - 1/2 g_ab R?

Einstein's choice of R_ab - 1/2 g_ab R is a mathematical equation that he used in his theory of general relativity. It is known as the Einstein field equations and it describes the relationship between matter and the curvature of space-time.

## Why did Einstein choose this specific equation?

Einstein chose this specific equation because it is a natural extension of his theory of special relativity. It takes into account the effects of gravity on the curvature of space-time, which was a major advancement in our understanding of the universe.

## What does R_ab - 1/2 g_ab R represent?

R_ab - 1/2 g_ab R represents the Ricci tensor (R_ab) minus half the scalar curvature (R) multiplied by the metric tensor (g_ab). This equation is used to describe the curvature of space-time caused by the presence of matter and energy.

## How does this equation relate to general relativity?

This equation is a key component of Einstein's theory of general relativity. It is used to calculate the curvature of space-time, which is then used to describe the behavior of objects in the presence of gravitational fields.

## Are there any real-world applications of this equation?

Yes, there are many real-world applications of this equation. It has been used to accurately predict the behavior of objects in space, such as the orbit of planets and the bending of light around massive objects. It is also used in the development of technologies like GPS and in the study of black holes and the origins of the universe.

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