# Tag Info

## Hot answers tagged metric-tensor

### What is the evidence that gravitational fields don't sum up as a superposition?

Black hole solutions would not exist in a linear theory of gravity. This is because black holes are vacuum solutions, not sourced by any matter, and there are no static vacuum solutions that die off ...
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### Where is the Lorentz signature enforced in general relativity?

The Lorentz signature is just part of the theory; for example in a weak-field limit, we should reduce to special relativity, which is described using Lorentz signature (in order to talk about light, ...
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### Physical significance of metric compatibility

There is a formulation of GR where metric compatibility follows from the equations of motion. If you think of the Einstein-Hilbert action as being a functional of both the connection and the metric (...
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### What is the evidence that gravitational fields don't sum up as a superposition?

Einstein's equations are $$G_{\mu\nu}[g] = R_{\mu\nu}[g] - \frac{1}{2} g_{\mu\nu}R[g] = 8\pi G_N T_{\mu\nu} \tag{1}.$$ where $g_{\mu\nu}$ is the metric of the spacetime. The Ricci scalar is given by ...
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### Where is the Lorentz signature enforced in general relativity?

As a simplified, purely mathematical answer, the signature of the metric is baked into the initial/boundary conditions. First of all note that it cannot change along a smooth enough connected space-...
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### Contravariance and covariance of vectors

That's because you're using the 'fake' version of gradient. The true version for a scalar field $F$ is given as: $$(\nabla F)= \frac{\partial F}{\partial Z^i } e^i$$ Here $e^i$ is the dual basis and ...
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### What is the evidence that gravitational fields don't sum up as a superposition?

Gravitational wave (GW) observations of binary black holes (BH) may provide experimental tests of superposition of spacetimes, as defined by you. Each BH is described by a spacetime metric, but the ...
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### Riemann curvature tensor in an inertial frame

The fact that a function's first derivative vanishes at a point does not mean that its second derivative vanishes at that point. Note that for $f(x)=x^2$, $f'(0)=0$ but $f''(0)=2$.
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### Could 2D spacetime be seen as an embedded manifold?

If you mean "embedded in $\mathbb{R}^3$ with a Euclidean metric", then the answer is no. Suppose that such an embedding exists. Consider the neighborhood of a point $P$ on the submanifold. ...
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### How do we figure out what is the right geometry of space?

Don't we we already have a geometry of space time as soon as we write down the Minkowski metric? Yes, but that's putting the cart before the horse. Einstein's equations are a system of differential ...
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### Are there types of spacetime that have no symmetries?

I assume you're actually asking about isometries of the metric $\phi^* g = g$ in the context of GR (physical spacetime symmetries). The 'gauge symmetry' of GR, diffeomorphism invariance, is an ...
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### Where is the Lorentz signature enforced in general relativity?

Quoted from Pseudo-Riemannian manifold - Lorentzian manifold: A principal premise of general relativity is that spacetime can be modeled as a 4-dimensional Lorentzian manifold of signature $(-,+,+,+)$...
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### Where is the Lorentz signature enforced in general relativity?

The Lorentzian signature of the metric is "baked into" the local causal structure (the set of "light cones", one at each event) of spacetime, which plays a role in the initial ...
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### Could time be a secondary effect due to curvature of space?

You can't have curvature in 1 dimension: it is a line and always locally flat, with $R^a{}_{bcd} = R^0{}_{000} =0$. So it doesn't make sense to ask about 'time being curved'. If you want to consider ...
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### Invariance of spacetime interval by Schutz

I would suggest you read Landau & Lifshitz argument for invariance of $ds^2$, particularly the Wikipedia link because it states clearly the theorem which is being proved, and it fills in a few ...
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### Linear Momentum in General Relativity

Is there a metric that describes this case? Yes. It is the Schwarzschild metric (valid outside of the gravitating body if we are talking about something like a star). When written in the form where ...
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