# Tagged Questions

Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.

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### The role of metric in the Wave Equation

The wave equation is often written in the form $$(\partial^2_t-\Delta)u=0,$$ involving the Laplace-Beltrami operator $\Delta$. However, the Laplace-Beltrami operator $\Delta$ is defined only in the ...
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### Can general relativity be explained by equations describing a fabric of space embedded in a flat 5-dimensional Minkowski space?

Does such a set of equations exist or does our universe have an intrinsic curvature that can't be explained by an embedding in a flat Minkowski space of 1 higher dimension? Even if general relativity ...
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### From Manifold to Manifold?

Tensor equations are supposed to stay invariant in form wrt coordinate transformations where the metric is preserved. It is important to take note of the fact that invariance in form of the tensor ...
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### D'Alembertian for a scalar field

I have read that the D'Alembertian for a scalar field is $$\Box = g^{\nu\mu}\nabla_\nu\nabla_\mu = \frac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}\partial^\mu).$$ Exactly when is this correct? Only for ...
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### Is $ds^2$ just a number or is it actually a quantity squared?

I originally thought $ds^2$ was the square of some number we call the spacetime interval. I thought this because Taylor and Wheeler treat it like the square of a quantity in their book Spacetime ...
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### Conservative vector fields

I was always told that to find whether or not a vector field is conservative, see if the curl is zero. I have now been told that just because the curl is zero does not necessarily mean it is ...
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### Why isn't invariant notation common?

In principle, one can write quantities in a manifestly invariant - rather than covariant - fashion in e.g. special relativity. For example, rather than writing just $x^\mu$, we could write the basis ...
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### Coordinate Transformation of Scalar Fields in QFT

By definition scalar fields are independent of coordinate system, thus I would expect a scalar field $\psi [x]$ would not change under the transformation $x^\mu \to x^\mu + \epsilon^\mu$. Correct? ...
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### In the static spacetime, the extrinsic curvature of hypersurface $t=constant$ is zero

How can I prove that in the static spacetime, the extrinsic curvature of hypersurface $t=constant$ is zero? My efforts all are failed. Any hint would be greatly appreciated.
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### Fourier Transform on a Riemannian Manifold

The question is quite simple: What would be the definition of Fourier Transform (and it's inverse) on a Riemannian Manifold? I've found that a similar question has been asked at Mathematics.SE but ...