Question

In 2D the Einstein tensor is always zero, and we can easily get solution with non-zero Ricci tensor but zero Einstein tensor. But is it possible in 4D? Can we get a space-time with zero Einstein tensor, zero cosmological constant but non-zero Ricci tensor and Ricci scalar? If not, why?

Answer 1

In 4D, it is generally not possible to have a space-time with zero Einstein tensor and non-zero Ricci tensor and Ricci scalar. The Einstein tensor represents the curvature of spacetime and its relation to matter and energy, and it is determined by the Ricci tensor, the Ricci scalar, and the metric tensor.

Step-by-step explanation:

In 2D, the Einstein tensor is always zero, but in 4D, it is generally not zero. The Einstein tensor is a combination of the Ricci tensor, Ricci scalar, and the metric tensor.

It represents the curvature of spacetime and its relation to matter and energy. In 4D, if the Ricci tensor and Ricci scalar are non-zero, it is not possible for the Einstein tensor to be zero, unless there is a cancellation between different terms in the equation.

This can be understood by considering the Einstein field equations, which relate the curvature of spacetime to the distribution of matter and energy:

Gμν = 8πTμν

where Gμν is the Einstein tensor, Tμν is the stress-energy tensor, and the constant 8π is related to the gravitational constant. If the Ricci tensor and Ricci scalar are non-zero, then the stress-energy tensor must also be non-zero in order to satisfy the equation. Therefore, it is not possible to have a space-time with zero Einstein tensor and non-zero Ricci tensor and Ricci scalar.