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

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Are diffeomorphisms a proper subgroup of conformal transformations?

The title sums it pretty much. Are all diffeomorphism transformations also conformal transformations? If the answer is that they are not, what are called the set of diffeomorphisms that are not ...
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0answers
32 views

Laplacian in 4 spatial dimensions; 4th dimension warped

How can I prove the form of the Laplacian in four spatial dimensions, using the identification $y = y + 2\pi R$ for the fourth dimension and assuming the others as the usual Cartesian ones? I want to ...
8
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2answers
314 views

AdS Space Boundary and Geodesics

I'm new to working with AdS space and am primarily concerned with black holes. I'm just playing round with the metric for AdS$_4$ $$ds^2=-f(r)dt^2+f^{-1}(r)dr^2+r^2d\zeta^2$$ for $f(r)=r^2+m $, ...
5
votes
3answers
121 views

Integral in different coordinate systems

In Griffiths' electrodynamics book, he uses the equation, $$\nabla^2\mathbf{A}=-\mu_0 \mathbf{J},$$ to state that $$\mathbf{A}(\mathbf{r}) = ...
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4answers
1k views

Why is the space-time interval squared?

The space-time interval equation is this: $$\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2-(c\Delta t)^2$$ Where, $\Delta x, \Delta y, \Delta z$ and $\Delta t$ represent the distances along various ...
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0answers
87 views

Simple General Relativity Relation [closed]

Given the identity $$\nabla_a(R^{ab}-\frac{1}{2}R g^{ab})=0,$$ how do I then show that $R_{ab}=0$ implies $$\nabla_a R^a_{\space \space \space bcd}=0$$
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4answers
185 views

Where does the idea gravity=curvature of spacetime really come from?

I have been searching for quite a while but mostly found the answer: Einstein's genius. Quite unsatisfactory. I know and understand that the idea gravity=curvature of spacetime works. Furthermore I ...
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0answers
54 views

Recommendation on mathematical physics book of Symplectic geometry

I want to learn the applications of symplectic geometry in physics. Which mathematical physics textbook will have a detailed and heuristic explanation of this aspect?
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68 views

How is foliation of manifolds' theory useful in General Relativity?

I am interested on getting some hints on how Foliations Theory of Manifolds can be used fruitfully on General Relativity. I just started my Ph.D on Mathematics this semester focusing on studying ...
3
votes
3answers
103 views

Linear independence of the Covariant Derivative

What's the easiest way to show that the covariant derivative $\nabla U^{\mu}$ is linearly independent to $U^{\mu}$, which is a vector? I mean I'm assuming they are since I'm proving the second ...
2
votes
1answer
100 views

Stuck following derivation of geodesic equation

In the book "Reflections on Relativity" by Kevin Brown, there is a chapter called "Relatively Straight", in which he derives the geodesic equations using the Euler equation. Online version Just ...
3
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1answer
217 views

Gregory-Laflamme Instability of Black Strings and $p$-Branes

In a paper by Gregory and Laflamme (http://arxiv.org/abs/hep-th/9301052) in 1993, it was demonstrated that black strings and $p$-branes which were solutions to certain low energy string theories were ...
5
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1answer
88 views

What is the relationship between Cyclic Coordinates and Killing Vector Fields?

My question is related to this question. There are three or four other questions on Killing Vector Fields here, however none of them that I have seen address my question. $\\$ I've been studying ...
2
votes
4answers
166 views

GR matter-free equations and Schwarzschild geometry

I am reading some lecture notes on General relativity (undergraduate level) and I do not understand a sequence of statements about the topics in the title. After stating that the for matter-free ...
1
vote
1answer
67 views

Knots and singularities

Can space-time singularities be treated as mathematical knots occurring in dimensions greater than four? I just drew an analogy with knots in one-dimensional strings. When a rubber-band is looped over ...
4
votes
1answer
59 views

What's the point of a cobasis?

I've been learning about tensor analysis, and things have been going well so far, but I'm a bit stuck when it comes to the idea of a cobasis (by which I mean the reciprocal basis; not sure which term ...
3
votes
1answer
109 views

Taylor expansion of the metric

Consider a coordinate change $$ x^a\mapsto \tilde x^a=x^a+\epsilon y^a $$ In the note I am reading, the author calculate the change of metric by $$ g_{ab}(x) = \tilde g_{ab}(\tilde x)=\tilde ...
5
votes
2answers
181 views

Confusion about Lie derivative on metric

According to this site, the Lie derivative of a $(0,2)$-tensor is $$ \mathcal{L}_XT_{ab}=\partial_XT_{ab}+T_{cb}\partial_aX^c+T_{ac}\partial_bX^c $$ However, according the same website, the Lie ...
2
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0answers
46 views

What is the Levi--Civita connection of a Wick rotated metric?

A Wick rotation is a transformation that allows to change from a Lorentzian manifold to a Riemaniann manifold. In the cases when this is possible, is the Levi-Civita connection of the Riemaniann ...
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0answers
67 views

About the proof of the second Bianchi Identity

The second Bianchi Identity is $$ \nabla_{[a}R_{bc]de}=0 $$ As far as I know, the proof (say, Walfram Mathword) start by stating the representation of Riemann tensor in local inertial coordinates $$ ...
13
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2answers
280 views

Global Properties of Spacetime Manifolds

When solving the Einstein field equations, $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi GT_{\mu\nu}$$ for a particular stress-energy tensor, we obtain the metric of the spacetime manifold, ...
2
votes
1answer
52 views

de Rham Cohomology of Schwarzschild Manifold

Let $C^p(M)$ denote the group of closed $p$-forms on the manifold $M$, and $Z^p(M)$ the group of all exact $p$-forms on the manifold $M$. The de Rham cohomology is given by the quotient, ...
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1answer
86 views

proper distance and proper length

I am wondering if I mix up the notion of proper distance and proper length. I have two cuves in Schwarzschild space-time describing the flight of two photons (think of it as photons guided in by ...
3
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2answers
263 views

Space-time Topologies?

When it comes to questions of existence of bounds for PDE's and such, one must often make some assumptions regarding the topology of the space-time to use well known theorems. My question is ...
4
votes
1answer
94 views

The properity of $\mathbb{R}^4$ that has infinitely many differential structures is related to Yang-Mills field?

I heard a saying that $\mathbb{R}^4$ having infinitely many differential structures which are not diffeomorphic to each other has a relationship with Yang-Mills field. Does anyone can explain it, and ...
11
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1answer
378 views

Physical Interpretation of EM Field Lagrangian

Using differential forms and their picture interpretations, I wonder if it's possible to give a nice geometric & physical motivation for the form of the Electromagnetic Lagrangian density? The ...
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2answers
50 views

Show that two families of curves are orthogonal (without using orthogonal trajectories)

I'm reading through Hartle's General Relativity and came across this question: Consider the following coordinate transformation from rectangular coordinates $(x,y)$, labeling points in the plane ...
2
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1answer
83 views

Calculate divergence of vector in curvilinear coordinates using the metric

In a curved $(3+1)$ dimensional spacetime with metric components $g_{\mu \nu}$, the covariant derivative of a $4$ vector $\mathbf V = (V^0, \vec V)$ is given by $$\nabla_\mu~ V^\mu = ...
2
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0answers
60 views

$M^{+}_4$ Randall-Sundrum Brane Calculation

The basic Randall-Sundrum model is given by the metric, $$\mathrm{d}s^2 = e^{-2|\sigma|}\left[ \mathrm{d}t^2 -\mathrm{d}x^2-\mathrm{d}y^2 - \mathrm{d}z^2 \right]-\mathrm{d}\sigma^2$$ where $\sigma$ ...
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0answers
151 views

Differential geometry of Lie groups

In Weinberg's Classical Solutions of Quantum Field Theory, he states whilst introducing homotopy that groups, such as $SU(2)$, may be endowed with the structure of a smooth manifold after which they ...
3
votes
1answer
101 views

What is the difference between $\nabla _{\sigma} $ and $ \nabla^{\sigma}$?

What is the difference between: $\nabla _{\sigma} $ and $ \nabla^{\sigma}$? I've been told that the first is the covariant derivative, however I'm just starting a course on spacetime geometry and ...
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vote
2answers
70 views

Coset space and transitiviy

I have a question regarding coset space or homogeneous space $SO(n+1)/SO(n)$ which is simply $S^n$. I need some intuition regarding this result. As everyone knows that for a simple case of ...
8
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1answer
115 views

Evaluating the Einstein-Hilbert action

The Einstein-Hilbert action is given by, $$I = \frac{1}{16\pi G} \int_{M} \mathrm{d}^d x \, \sqrt{-g} \, R \, \, + \, \, \frac{1}{8\pi G}\int_{\partial M} \mathrm{d}^{d-1}x \, \sqrt{-h} \, K$$ ...
2
votes
1answer
74 views

Are covariant derivatives of Killing vector fields symmetric?

I'm reading the Lecture Notes on General Relativity by Matthias Blau, and in section 9.1 (point 1) he writes: Let $K^\mu$ be a Killing vector field, and ${x^\mu(\tau)}$ be a geodesic. Then the ...
3
votes
3answers
326 views

Why Hausdorff and Paracompact manifold in GR?

What can we say about the transition map if the manifold is a Hausdorff space? Why do we need the manifolds to be Hausdorff and paracompact in General Relativity?
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0answers
108 views

Why do we need non-trivial fibrations?

I am currently reading this paper. I understand how the Bloch sphere $S^2$ is presented as a geometric representation of the observables of a two-state system: $$ \alpha |0\rangle + \beta |1\rangle ...
2
votes
1answer
143 views

Spin connection in higher dimension

I have a problem regarding computation of spin connection in the case where One or more dimension is compactified. For example if we take a $D+1$ dimensional bosonic string action and write the $D+1$ ...
3
votes
1answer
58 views

Which textbook of differential geometry will introduce conformal transformation?

Which textbook of differerntial geometry will have these formulas about conformal transformation? $$\tilde g_{ij} = e^{2\varphi}g_{ij}$$ $$\tilde \Gamma^k{}_{ij} = \Gamma^k{}_{ij}+ ...
2
votes
2answers
136 views

What is the meaning of space-time curvature?

What is the difference between the Space-time curvature and Space curvature?
3
votes
1answer
90 views

Ricci tensor of direct product of manifolds

Imagine I have a (Lorentzian) manifold with a metric $\left[ {\begin{array}{cc} g_{\mu\nu} &0\\ 0&g_{mn}\\ \end{array} } \right]$ Will the Ricci tensor be also block diagonal ...
4
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0answers
77 views

Why is the Ricci tensor diagonal for isotropic spacetime?

I'm reading Zee's Einstein Gravity in a Nutshell and while calculating the Ricci tensor for FRW spacetime he claims that because the spacelike slices of constant $t$ are rotationally invariant, the ...
4
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0answers
66 views

Intuitively what's the relationship between forces and connections?

In Einstein's General Relativity we relate the effects of gravity with the curvature of the Levi-Civita connection on the spacetime manifold. Also, when we get the electromagnetic tensor $F = dA$ ...
2
votes
1answer
144 views

The Spin Connection

Why do we need to introduce the spin connection coefficients $\omega_{\mu \space \space b}^{\space \space a} $ in General Relativity? To me, they just look (mathematically) like the Christoffel ...
3
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0answers
49 views

Rigid rectangle in Schwarzschild

Say I build a perfect rectangle. Side lengths $l_1$ and $l_2$ and perfect right angles. I am on earth and the metric is given by the Schwarzschild metric. Setting $dt=0$ leads to the spatial ...
1
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1answer
87 views

Metric for infinite straight cosmic string

A string theory question on my general relativity problem set: Metric is given as $$\mathrm{d}s^2 = -A(r)\mathrm{d}t^2 + B(r)\mathrm{d}r^2 + r^2 \mathrm{d}\theta^2.$$ a) Solve the vacuum equations ...
4
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2answers
116 views

What is the notion of a spatial angle in general relativity?

Is there a notion of spatial angles in general relativity? Example: The world line of a photon is given by $x^{\mu}(\lambda)$. Suppose it flies into my lab where I have a mirror. I align the mirror ...
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1answer
77 views

How to calculate spatial distance in space-time?

Pinning two test particles at two different points in space, how can I calculate their spatial distance, when the geometry is given by the Schwarzschild metric? Let's say particle 1 is pinned at ...
0
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1answer
61 views

Covariant derivative of a vanishing tensor component [closed]

Is the covariant derivative of a vanishing tensor component necessarily zero?
0
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1answer
87 views

Geodesic devation on a two sphere

So the geodesic deviation equation gives the relative acceleration between two geodesics in motion. But given a pair of geodesic (let's say on the two sphere) that start at the equator, separated by ...
4
votes
3answers
91 views

Aside from experimental evidence, is there any reason to model space as Euclidean?

Obviously experiment is the end-all-be-all of any science, but I'm curious if there's any a priori reason to model space as Euclidean three-space (from a pre-relativity viewpoint, of course; I'm ...