# Questions tagged [differential-geometry]

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

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### Conserved quantities with charge and Killing's vectors

I'm trying to solve the following problem: A particle with electric charge e moves with 4-velocity $U_{\alpha}$ in a spacetime with metric $g_{αβ}$ in the presence of a vector potential $A_µ$. The ...
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### Nambu-Goto action and the World-Sheet Area

I am studying string theory from the book "String theory and M-theory", written by Becker, Becker and Schwartz. My question is: We are told that the Nambu-Goto action is simply the one that ...
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### Metric Tensor times its inverse (non-zero curvature)

so I am quite confused regarding the spatial metric tensor $g_{ij}$. If I have $g_{ij}g^{ij}$ I essentially get the trace of the metric tensor $g$ right? Or, do I get $\delta^i_i = 3$ instead? The ...
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### Confusion regarding Riemann Tensor and Ricci Tensor

Ricci Tensor is the contraction of the Riemann Tensor. Even if all the components of the Ricci Tensor is zero, that doesn't mean that the spacetime is flat. If all the components of the Riemann Tensor ...
1 vote
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Suppose an affine parameter $\lambda$ is defined along a null geodesic with $dx^\mu/d\lambda=k^\mu$. How could I write the partial derivative $\partial f/\partial x^\mu$ by using $df/d\lambda$? If $k^\... 4 votes 2 answers 149 views ### Lie group generators - exponential map In the framework of Quantum Physics, I have to explain to some of my colleagues what is a Lie group, a Lie algebra and their connections with the exponential map. This is mainly to make them ... • 51 0 votes 1 answer 44 views ### How can I calculate the solid angle of a planet? [closed] I've read several articles about solid angles recently and have one question now. They can be calculating by using the following formula:$\Omega = \frac{A}{r²}$A stand for the area and r for the ... 1 vote 0 answers 127 views +50 ### HaMiDeW coefficients - recursive calculation of the coincidence limits In his book Aspects of Quantum Field Theory in Curved Spacetime Stephen Fulling calculates the coincidence limit$[a_1]$and gives an idea of how$[a_n]$with$2 ≤ n$can be found recursively. Since ... 0 votes 0 answers 59 views ### What is the$r$coordinate in a$\mathbb{S}^3$FLRW spactime? I'm having trouble understanding what the$r$reduced-circumference coordinate really is in a 3-sphere$\mathbb{S}^3$context. Let's start with the unit 3-sphere metric in hyperspherical$(\psi, \...
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Is this definition of Lorentz transformations correct? Consider 3+1 dimensional space-time manifold $M$. Let $v,u$ are two vectors of the vector bundle and $g$ be a metric on $M$. Now we can define ...
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### What diffeomorphism does the Hamiltonian constraint generate?

Consider the Hamiltonian constraint $\mathcal H(x)$ in the ADM formalism. What diffeomorphism does this generate?
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### Partial derivative of Christoffel symbols

I have faced one issue regarding the partial derivative of the Christoffel symbols. $\Gamma$'s themselves are not tensors. But if we take the difference between two $\Gamma$'s at the same point, then ...
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### Lie algebra of a Lie group

I'm going through some notes on group theory for physics. After introducing the concept of Lie group and Lie algebra the writer makes the connection between the two. Let $G$ be a Lie group of ...
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### Relation between Lorentz transformations in QFT and GR [duplicate]

I often have difficulty expressing certain doubts because I am not (not even my self, yes) fully aware of what's going on that bothers me, so forgive me if the question isn't the clearest. I noticed ...
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