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

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In Schutz's A First Course in General Relativity (p122) he derives the polar coordinate basis vector$$\vec{e_{r}}=\frac{\partial x}{\partial r}\vec{e_{x}}+\frac{\partial y}{\partial r}\vec{e_{y}.}$$ …
asked Jan 15 '16 by Peter4075
3
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1answer
In the context of four-dimensional spacetime, how does the metric turn a tangent vector into a gradient, and vice versa? By this I mean that I know the metric can be used to raise and lower indices: …
asked Sep 9 '13 by Peter4075
6
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1answer
I've calculated the correct answer to my problem, but don't understand one of the assumptions I made when doing so. I used the geodesic deviation equation $$\frac{D^{2}\xi^{\mu}}{D\lambda^{2}}+R_{\ph …
asked Oct 17 '14 by Peter4075
3
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3answers
In the context of GTR spacetime, I'm trying to get the basic idea of a Riemannian manifold clear in my mind. Apologies for the longwindedness. Question 1. Is this a reasonable, simplified summary of …
asked Oct 21 '11 by Peter4075
1
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1answer
I don't understand the derivation of Equation 2.14$$\mathrm{d}f\left(\frac{d}{d\lambda}\right)=\frac{df}{d\lambda} \tag{2.14}$$ in Carroll's Lecture Notes on General Relativity (http://ned.ipac.calte …
asked Oct 25 '15 by Peter4075
2
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1answer
Very little interest in the original version of this question so I've rejigged it hoping for a more positive response. I'm trying to use the geodesic deviation equation$$\frac{D^{2}\xi^{\mu}}{D\lambd …
asked Sep 3 '14 by Peter4075
5
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This is my non-physicists take. A manifold is a curved space that is locally flat. Think of the surface of the Earth, which is a two-dimensional manifold (can be described using two coordinates - lati …
answered Dec 11 '16 by Peter4075
3
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1answer
Trying to work through a textbook derivation of the geodesic deviation equation, I've calculated this identity:$$u_{;\beta}^{\alpha}u_{\alpha}=u_{\alpha;\beta}u^{\alpha}.$$ If this is true, I'm maki …
asked Nov 17 '14 by Peter4075
1
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1answer
In the context of spacetime, reading Schutz, I'm confused about the symmetries of the Riemann curvature tensor, which I understand are: $$R_{\alpha\beta\gamma\mu}=-R_{\beta\alpha\gamma\mu}=-R_{\alpha\ …
asked Dec 18 '13 by Peter4075
2
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1answer
I'm stuck trying to follow Foster and Nightingale's derivation of the geodesic equation from two neighbouring geodesics $x^{a}\left(u\right)$ and $\tilde{x}^{a}\left(u\right)$ joined by a connecti …
asked Aug 28 '15 by Peter4075
3
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I've since found out that where I'm going wrong is that I don't need to find the absolute derivative. I've been told on another physics forum that this problem is framed in terms of Riemann normal coo …
answered Oct 1 '14 by Peter4075
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2answers
My previous question Textbook disagreement on geodesic deviation on a 2-sphere got shot down as “off topic”, so I'm having a second stab at it. Misner et al's Gravitation (p34) gives the geodesic de …
asked Aug 28 '14 by Peter4075
2
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3answers
I'm trying to understand the derivation of the Riemann curvature tensor as given in Foster and Nightingale's A Short Course In General Relativity, p. 102. They start by giving the covariant derivative …
asked Nov 21 '14 by Peter4075
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0answers
Apologies if I have this completely wrong (and for the general long-windedness). I've searched online but can't find anything helpful/relevant. I'm trying to use the geodesic equation $$\frac{D^{2} …
asked Aug 27 '14 by Peter4075
5
votes
1answer
I'm trying (very early stages) to understand the derivation of the geodesic equation $$\frac{d^{2}x^{\alpha}}{d\lambda^{2}}+\Gamma_{\gamma\beta}^{\alpha}\frac{dx^{\beta}}{d\lambda}\frac{dx^{\gamma}}{ …
asked May 12 '15 by Peter4075

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