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Large masses in space as stars and planets cause a curvature in the spacetime fabric. What are the factors that affect this curvature? Is it only mass? And can we conclude these factors using Tensors?

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The Einstein field equations which relate physical quantities to curvature are:

$$R_{\mu \nu} -\frac{1}{2}g_{\mu \nu} R = 8\pi G T_{\mu \nu}$$

The tensor which contributes to the curvature is the stress-energy tensor $T_{\mu \nu}$ which contains quantities such as energy density, shear stress and pressure. The tensor itself is computed from the Lagrangian which governs the matter present. In field theories, we see via Noether's theorem that the stress-energy tensor is in fact the conserved current due to invariance (up to a total derivative) under global spacetime translations.

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The curvature is determined by the stress-energy tensor. Mass (or more precisely energy density) contributes to this, but so do momentum density, shear stress and pressure.

The Einstein equation is a tensor equation that relates the curvature tensor to the stress-energy tensor.

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