# 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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### Is this equation $\nabla_a\sqrt{-g}=0$ correct? [duplicate]

Is the equation $$\nabla_a\sqrt{-g}=0$$ correct? Here $\nabla_a$ is the Levi-Civita connection, and $g$ is the determinant of metric $g_{ab}$. Apparently, we have $\nabla_ag_{bc}=0$, but I am not sure ...
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### Conformal Transformation: Minkowski sheet to cylinder

What conformal transformation can I make to 2d Minkowski with metric $ds^2=-dt^2+dx^2$ to show that it is conformal to a cylinder?
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### What is the Metric Tensor? [duplicate]

I was studying Einstein's Field Equation, and this was the most common symbol. Can you explain what it is and how it could be used?
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### Conformal Connections in Physics [closed]

For my diploma thesis in Mathematics I investigate conformal connections (as an example of Cartan connections). All in all the thesis should deal with geometric aspects (associated bundles, (pseudo)-...
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### When and where to check the formal definition of a manifold

In most texts on GR we are first introduced to a formal and rigorous definition of a manifold. We then learn the point that in GR "any coordinate system" might be used for the 4D spacetime metric. ...
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### Why is the covariant derivative of the determinant of the metric zero?

This question, metric determinant and its partial and covariant derivative, seems to indicate $$\nabla_a \sqrt{g}=0.$$ Why is this the case? I've always learned that $$\nabla_a f= \partial_a f,$$ ...
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### Straight lines in general relativity

This question stems from a possibly misguided attempt to understand General Relativity. I am about to leave High school for college, I do however have a rudimentary understanding of tensors, and I ...
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### Continuity Equation in Differential Geometry

I'm looking for a derivation of the mass continuity that applies in general on symplectic manifolds. In particular the "the amount of change in the mass in a volume is just amount that flows in or out"...
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### Books about non-euclidean geometry [duplicate]

I study physics and want to know the best books to learn non-euclidean geometry in an easy way.
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### Acceleration of particle “held in place” at $x = 1$ [closed]

The metric components in a two-dimensional spacetime are given in terms of the coordinates $(t, x)$ by$$ds^2 = -\cosh x\,dt^2 + dx^2.$$Consider a particle that is "held in position" at $x = 1$. What ...
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### What's the difference between the diffeomorphism invariance and reparametrization invariance?

Can somebody tell me what's the difference between the diffeomorphism invariance and reparametrization invariance?
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### Equivariant cohomology and Mayer-Vietoris sequence

I'm reading this article upon topological field theory and I'm a bit confused about the way he compute equivariant cohomology of $S^2$ wrt $\mathrm{U}(1)$, i.e. $H^\bullet_{S^1}(S^2)$. You can find ...
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### GR - curve (in)completeness & (in)extendibility

Seeking clarification of the distinction between completeness of geodesics/extendibility of curves in GR spacetimes? (Confirm: not the geodesic completeness of a spacetime but the completeness of an ...
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### Two “Robertson-Walker observers,” velocity of baseball as seen by second observer right before it's caught?

The spacetime metric of a spatially flat ($k = 0$) radiation dominated FLRW universe is given by$$ds^2 = -dT^2 + T[dx^2 + dy^2 + dz^2].$$Consider two "Robertson-Walker observers," i.e., observers with ...
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### Spacetime manifold surgery: is this result still a valid etc. spacetime?

Given a valid classical GR spacetime manifold $M$ (i.e. 4D, Lorentzian, Hausdorff, paracompact, ?etc.), and $B\subset M$, a closed spatial subset (e.g. a closed ball at fixed $t$) to be excised, [...
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### Book recommendations on geometrical methods for physicists (like Topology, Diff. Geometry) [duplicate]

I would like to obtain a book that has to do with geometrical methods/subjects for physicists. When I say geometrical methods/subjects I mean things like Topology, Differential Geometry, Lie ...
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### probability of striking the circular ring by gas molecules

In kinetic theory we use probabilistic case to derive pressure, no. Of molecules having speed c to c+dc or in such cases.and to derive such equations we introduce a term called "SOLID ANGLE" I come ...
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### If we live on the surface of Earth then why Earth images shows maps around it? [closed]

If you visits google map and go to earth we see the image as attached below. My question is if the earth is round like sphere ball and if we live on the surface of this ball (point me if i am ...
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### Why do we need connections, if we have the Lie derivative?

When I learned about the covariant derivative, it was motivated as a way of defining a good differentiation operation on tensors. To do this, we had to define a connection on the manifold, which was a ...
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### What do we exactly mean by a “topological object” in physics?

I have been working on topological defects like monopoles, etc. for some time. One think that I have not been able to understand is the physical meaning of the phrase "topological object". I have ...
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### Extrinsic curvature components

I'm trying to understand how to derive the extrinsic curvature (in order to understand some calculation on fluid/gravity dynamics). But I hit a wall in my progress. I stuck at trying to verify the ...
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### How do we measure Schwarzschild coordinates?

In special relativity, we make a big fuss about setting up inertial frames of reference, and then constructing coordinate systems using networks of clocks and rulers. This gives an unambiguous ...
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### A question regarding $f(R)$ Lagrangians

Consider the class of Lagrangian known as $f(R)$ Lagrangians where the Lagrangian is some function $f(R)$, $$S=\int\sqrt{g}d^4x\ f(R)$$ assuming there are no (or ignoring)...
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### Complex tetrad vs. Real metric

I asked this question almost a month ago on mathoverflow (http://mathoverflow.net/q/228138/) but received no response. I thought I could have better luck here: I have a question on the relationship ...
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### Is it possible to integrate a function over a null hypersurface?

For $f$ a compactly supported function on a spacetime $(\mathcal{M},g)$, one may define its integral over $\mathcal{M}$ by $$f\longmapsto \int_\mathcal{M}\star f,$$ where $\star$ is the Hodge star of ...
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### How do you know what kind of space(time) you have when solving the Einstein Field Equations?

I'm experimenting with the EFE, and I ''invented'' a metric; a diagonal non-zero metric, and I discovered that the Riemann tensors are equal to zero which implies the Einstein tensor $G_{mn}$ equals ...