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PerpStudent
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In the Einstein tensor equation for general relativity, why are there two terms for curvature: specifically the curvature tensor and the curvature scalar multiplied by the metric tensor?
When the Einstein-Hilbert action is extremized wrt the inverse metric, that is what emerges. See here http://en.wikipedia.org/wiki/Einstein–Hilbert_action.PerpStudent said:In the Einstein tensor equation for general relativity, why are there two terms for curvature: specifically the curvature tensor and the curvature scalar multiplied by the metric tensor?
PerpStudent said:In the Einstein tensor equation for general relativity, why are there two terms for curvature: specifically the curvature tensor and the curvature scalar multiplied by the metric tensor?
Ben Niehoff said:The "reason" the particular combination
[tex]G^{\mu\nu} \equiv R^{\mu\nu} - \frac12 R g^{\mu\nu}[/tex]
appears is because is this the unique combination of curvatures that satisfies
[tex]\nabla_\mu G^{\mu\nu} = 0.[/tex]
tiny-tim said:pervect, ben, as a matter of interest, do you know any easy-to-understand reason why it's 8π ?
4π I'm more or less used to (and even 4π 10-7 ) …
but why 8 ?
pervect said:To expand on this , Einstein's equation says that G_uv = 8 pi T_uv, where T_uv is the stress-energy tensor.
Continuiity conditions on the stress-energy tensor, T_uv require that
[tex]\nabla_\mu T^{\mu\nu} = 0.[/tex] i.e. that the tensor be divergence free.
So since T_uv, the rhs is divergence free, the lhs has to be divergence free as well.
tiny-tim said:from http://en.wikipedia.org/wiki/Stress–energy_tensor#Conservation_law …
The stress–energy tensor is the conserved Noether current associated with spacetime translations.
PerpStudent said:Is the requirement that [tex]\nabla_\mu T^{\mu\nu} = 0.[/tex] due to energy and momentum conservation?
PerpStudent said:Is the requirement that [tex]\nabla_\mu T^{\mu\nu} = 0.[/tex] due to energy and momentum conservation?
The curvature tensor, also known as the Riemann curvature tensor, is a mathematical object that describes the curvature of a manifold, which is a geometric space that can be curved. It is represented by a set of numbers that measure how much the space is curved at each point.
The existence of a curvature tensor is a consequence of the theory of general relativity, which states that gravity is not a force between masses, but rather a curvature of spacetime caused by the presence of mass and energy. The curvature tensor is used to describe this curvature and its effects on the motion of objects.
The curvature tensor is essential in understanding the behavior of gravity and the structure of the universe. It allows us to make predictions about the motion of objects, the bending of light, and the evolution of the universe. It also plays a crucial role in the mathematical formulation of general relativity.
The curvature scalar, also known as the Ricci scalar, is a mathematical quantity that is derived from the curvature tensor. It is obtained by contracting (summing) the components of the curvature tensor and represents the overall curvature of a space at a given point. It is a crucial parameter in the equations of general relativity.
The existence of the curvature scalar is a consequence of the mathematical properties of the curvature tensor. It provides a way to summarize the local curvature of a space in a single value, making it easier to analyze and compare different spaces. It is also used in the Einstein field equations to describe the relationship between the curvature of space and the distribution of matter and energy.