Do the field equations themselves constrain the metric tensor? or do they just translate external constraints on the stress-energy tensor into constraints on the metric tensor?(adsbygoogle = window.adsbygoogle || []).push({});

another way to ask the question is, if I generated an arbitrary differentiable metric tensor field, would it translate through the field equations to some stress-energy tensor field (and Weyl tensor field) that are physically valid?

The reason I am confused is because, while the field equation implies a continuity constraint (the covariant divergence is zero) and that constraint provides several important conservation laws, this constraint comes from a modification of the Bianchi identity, which itself just represents a symmetry of the Riemann curvature tensor, which exists for any differentiable metric field. So I don't see how any metric tensor field could ever breach the continuity constraint.

The same goes for the equation that constrains the derivative of the Weyl tensor, it is based on the Bianchi identity, which should be a property of (rather than a constraint on) the Riemannian curvature tensor.

But the metric tensor must be constrained, otherwise I'm sure it would be easy to generate a metric tensor field that involved a mass appearing out of nowhere, so breaching the conservation of mass.

Many thanks

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# I What constrains the metric tensor field in GR?

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