# Tagged Questions

Questions related to the mathematical aspects of Einstein's theory of relativity. For the physics and its interpretations, please ask at the physics.SE. You may also consider the tags (differential-geometry) and (pde).

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### Covariant Taylor series

I am reading the following lecture notes of Avramidi https://www.researchgate.net/publication/255565392_Analytic_and_geometric_methods_for_heat_kernel_applications_in_finance I want to understand ...
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### What do you need to understand the Theory of Relativity?

If someone has studied calculus, what "instruments" or what fields does one still need to understand the formulas behind the 2 theories of relativity (special and general)? By understand I mean more ...
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### Covariant derivative of parallel transport

I am learning Riemannian geometry and don't get why the following is true. We are on a Riemannian manifold with the Levi Cevita connection $\nabla$. Let $\mathcal{P}(x,x')$ be the parallel transport ...
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### Differential Geometry for General Relativity

I'm going to start self-studying General Relativity from Sean Caroll's Spacetime and Geometry: An Introduction to General Relativity. I'd like to have a textbook on Differential Geometry/Calculus on ...
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### What is the difference between intrinsic and extrinsic curvature?

In general relativity, energy bends spacetime. However, this doesn't mean that a fifth dimension for spacetime to "bend into" exists." That is, spacetime isn't embedded in a higher dimensional space, ...
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### What is the simplest way to describe a mathematical space?

As a complete noob in mathematics, I was wondering, what is the simplest way to describe a (preferably 2-dimensional, becuase it will be simpler) non-euclidean space in mathematics. For example in ...
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### Deriving E=mc^2 from hollow box with mass M and photon

I'm working on a problem to derive E=mc^2 using conservation of momentum and center of mass. We have a hollow block of length L and mass M. A photon passes through taking mass m and adding it to the ...
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### Confusion with conclusion to positive mass theorem

I am trying to understand the positive mass theorem as it is presented in the survey paper by Corvino and Pollack http://arxiv.org/abs/1102.5050 I am fundamentally confused by the structure of their ...
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### Gauge condition equivalent to condition that coordinate functions satisfy wave equation to first order

Let $\eta_{ab}$ be the metric of special relativity and let $x^\mu$ be global inertial coordinates of $\eta_{ab}$. Let $\gamma_{ab}$ be a small perturbation of $\eta_{ab}$. How do I see that the gauge ...
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### Confusion with abstract tensor notation

I am currently going through the abstract tensor notation in Wald's "General Relativity". I understand the purpose of it, but I need help understanding some of the conventions and definitions. So, ...
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### Intuition about antisymmetrizing tensor equations

I was looking at the symmetries of the Riemann tensor, and tried to prove a couple of properties, namely If $\nabla$ is torsion-free, then: (i) $R^a_{\,[bcd]}=0$, and (ii) $R^a_{\,b[cd;e]}=0$. ...
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### Motivation for Non-Euclidean geometry: relativity

I'm looking for references to motivate the study of non-Euclidean geometry. In particular I would like something about relativity. I do not want texts to learn non-Euclidean geometry, only ...
I was reading: https://en.wikipedia.org/wiki/Metric_tensor#Examples Is it correct that in the polar coordinate example, just after the euclidean metric example, that distance is measured as: $$\... 1answer 50 views ### Kerr spacetime not symmetric? I always see a term dt \, d \phi in the Kerr-spacetime. Now assuming this means dt \otimes d \phi this means that the Kerr spacetime is NOT(!) symmetric which is somehow non-sense. So do ... 1answer 56 views ### What is a Killing tensor? Wikipedia gives the definition of a Killing tensor. Unfortunately, I don't know how to interpret the parentheses (it is also not explicitly explained in the link) and was therefore wondering whether ... 0answers 39 views ### Does the metric in the Theory of Relativity actually satisfy the definition of a metric? Allow me to give a brief introduction to the topic, which has to do with physics; my question will still be a mathematical one. I think my question is aimed at people with a background both in physics ... 0answers 47 views ### Calculation of extrinsic curvature I asked this question first on physics.SE but I got no complete answer so I thought maybe someone here could help. I'm trying to understand how to derive the extrinsic curvature (in order to ... 1answer 30 views ### Einstein Summation Convention Minkowski Metric Picked up a book on General Relativity for Mathematicians, but I'm a bit unclear on some of the tensor notation. For example, the Minkowski Metric$$\eta_{\mu \nu} (\Delta x^\mu)(\Delta x^\nu)$$... 0answers 23 views ### Lifting the Einstein-Hilbert action into the frame bundle If we have a four dimensional real spacetime (M,g), with g being a (-+++) signature Lorentz-metric, and \{\theta^0,\theta^1,\theta^2,\theta^3\} is a local orthornormal coframe defined in some ... 1answer 67 views ### Calculating Christoffel symbols using variational geodesic equation Given the line element$$ds^2 = e^v dt^2 - e^{\lambda} dr^2 - r^2 d \theta^2 - r^2 \sin^2 \theta d \phi^2$$we wish to compute the Christoffel symbols \Gamma^{a}_{bc} using the geodesic equation. ... 2answers 112 views ### Where should the Lorentz transformations fit into this? I am trying to figure out how to "see" things in relativity via a toy model. With a pinhole camera I'd like to capture a relativistic scene consisting of a vertical marked stick which is moving ... 1answer 37 views ### Components of Maxwell tensor under Lorentz boost transformation The following is taken from exercise 12.4 in D'Inverno. We wish to compute the transformation properties of the electric field and magnetic induction under a Lorentz boost. Given the following boost ... 1answer 34 views ### Metric defining an sphere I want to find for which cases this metric can define an sphere:$$\frac{1}{P^2}\left(\mathrm d\theta^2+\sin^2 \theta\; \mathrm d\phi^2\right) where $P=\sin^2 \theta+K\cos^2 \theta$, with $K$ the ...
I've been trying to go through an example for parallel transport but I cannot quite follow the solution. A surface (paraboloid) is given by the parametric equation $r(ρ, φ)$ = $ρ \cos(φ)\hat{i}$ + \$ρ ...