# The Einstein and Ricci tensors

• TrickyDicky
In summary, Einstein decided to use the Einstein tensor instead of the Ricci tensor in the general relativity equations because it has zero divergence, making it a good choice for equating with the stress-energy tensor. This ensures local conservation of energy in the theory. While other divergence-free curvature tensors may have also been considered, the Einstein tensor was chosen for its ability to fulfill this requirement.
TrickyDicky
I'm trying to understand the Einstein field equations conceptually, and one of the things that I'd like to understand is why Einstein decided that the left side of the GR equation should be the Einstein tensor instead of the Ricci tensor, I heard that initially he entertained the idea of equating the Ricci tensor with the stress-energy tensor part, but he finally, in November 1915 came up with the final form. What were the physical reasons to decide that the Ricci tensor alone couldn't account for the curvature of the stress-energy tensor gravitational field?

Thanks

Because energy is conserved, the stress energy tensor has zero divergence. Since he wanted to equate the stress-energy tensor on the right-hand side with the curvature of space-time on the left-hand side, he needed a tensor on the left-hand side which also had zero divergence. The Ricci tensor has a non-zero divergence, so he created the Einstein tensor, which does have zero divergence.

phyzguy said:
Because energy is conserved, the stress energy tensor has zero divergence. Since he wanted to equate the stress-energy tensor on the right-hand side with the curvature of space-time on the left-hand side, he needed a tensor on the left-hand side which also had zero divergence. The Ricci tensor has a non-zero divergence, so he created the Einstein tensor, which does have zero divergence.

Thanks, I suspected it was related to energy conservation issues.
Can you give a an explanation for laymen of what it means to have zero divergence, and define divergence in this context?
Thanks.

Zero divergence means locally conserved. Like a fluid. That the quantity flowing into any small volume is equal to the quantity flowing back out of the volume.

In differential geometry, the Bianchi identity says that what we now call the Einstein tensor is a quantity for which the divergence is exactly zero always. By making a law equating this to the mass-energy tensor, you automatically ensure the mass-energy will be locally conserved according to the theory. But I assume other candidate divergence-free curvature tensors exist; there was probably further motivation for choosing the Einstein tensor specifically..

## What is the Einstein tensor?

The Einstein tensor, also known as the Einstein curvature tensor, is a mathematical object that describes the curvature of space-time in Einstein's theory of general relativity. It is a symmetric rank-2 tensor that combines the curvature of space-time with the distribution of matter and energy.

## What is the Ricci tensor?

The Ricci tensor, named after mathematician Gregorio Ricci-Curbastro, is a mathematical object that describes the intrinsic curvature of a Riemannian manifold (a mathematical space with a defined metric). In general relativity, it is used in conjunction with the Einstein tensor to describe the curvature of space-time.

## How are the Einstein and Ricci tensors related?

The Einstein tensor is a combination of the Ricci tensor and the scalar curvature (a measure of the overall curvature of a space-time). Specifically, it is the difference between the Ricci tensor and half of the scalar curvature. This relationship is described by Einstein's famous field equations in general relativity.

## What does the Einstein tensor tell us about space-time?

The Einstein tensor is a mathematical representation of the curvature of space-time, which is caused by the presence of matter and energy. It tells us how the geometry of space-time is affected by the distribution of matter and energy, and how this curvature determines the motion of objects in space-time.

## Why are the Einstein and Ricci tensors important?

The Einstein and Ricci tensors are crucial components in Einstein's theory of general relativity, which is our current understanding of gravity. They allow us to mathematically describe the curvature of space-time and how it is influenced by the presence of matter and energy. This has led to numerous predictions and explanations of phenomena such as the bending of light, the existence of black holes, and the expansion of the universe.

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