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

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What does coordinate invariance mean?

I would like to really understand what the mathematical as well as Physical meaning of coordinate invariance is. I have pretended to know what this means, but upon thinking a little harder today, I am ...
2
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2answers
188 views

What is the metric tensor for?

I am wondering how to use the metric tensor, in practice? I read the book and done the exercises in A student's guide to vectors and tensors by Dan Fleisch. The concept of a tensor and their ...
3
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1answer
106 views

Einstein tensor of a gravitational source

In section 4.4 of gravitational radiation chapter in Wald's general relativity, eq.4.4.49 shows the far-field generated by a variable mass quadrupole: $$ \gamma_{\mu \nu}(t,r)=\frac{2}{3R} \frac{d^2 ...
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1answer
111 views

Derivation of Schwarzschild metric using the full machinery of differential geometry [closed]

How would one derive the Schwarzschild metric using the full machinery of differential geometry, using the component approach as little as possible? Something along these lines: Begin with a manifold ...
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1answer
32 views

Inverse gauge transformation in general relativity [closed]

Can someone explain to me how (8.21) follows from (8.20). The Picture comes from A first course in general relativity (Schutz). Thanks and regards, Jens Wagemaker
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1answer
112 views

Why pseudo-Riemannian metric cannot define a topology?

It is not clear for me why a positive definite metric is necessary to define a topology as noted in some textbooks like the one by Carroll. Does this imply that in cosmology, say through FLRW metric, ...
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2answers
234 views

Symmetry of the Polyakov action?

Let us look at the Polyakov action for a string moving in a spacetime with metric $g_{\mu \nu}(X)$:$$S_P = -{1\over{4\pi \alpha'}} \int d^2 \sigma \sqrt{-\gamma} \gamma^{ab} \partial_a X^\mu ...
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1answer
65 views

Is an event formally a 4-vector? [duplicate]

An event is a 4D point in spacetime. At every point in spacetime there is a tangent space. 4-vectors live in the tangent space. One can contract two events using a metric tensor. Is there a process ...
2
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1answer
83 views

Killing vector and one-form [closed]

p. 21 in this paper (http://arxiv.org/abs/0704.0247) $V$ is Killing vector, where $V^2 = −4b\bar{b}$, which means it is timelike Killing vector. The authors say: From $V^2 = −4|b|^2$ and $V = ...
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0answers
50 views

Squashed spheres in general dimension

The point of this question is to help me find references regarding squashed spheres in general dimension. I am interested in the general theory of squashing for arbitrary dimension. All of the ...
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1answer
67 views

Variation of a tensor

Let a change of coordinates be given by $x^{\mu}\to x^{\mu '}=x^{\mu}+\varepsilon \xi^{\mu}(x)$ with epsilon a small quantity. Given a tensor $T$ we define $\delta T:=T'(x)-T(x)$. I guess this means ...
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2answers
103 views

Killing field in Minkowski space-time

If we look at the killing equation for a vector field $X$ in $\mathbb{R}^{(p,q)}$ (or on an open subset thereof) in coordinates with constant diagonal pseudo-metric we get: ...
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3answers
169 views

What is the physical meaning of the Levi-Civita connection?

I'm taking a course in General Relativity and I have studied the fundamental theorem of Riemannian geometry: Let $M$ be a manifold with metric $g$. Then exists an unique torsion-free connection ...
3
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1answer
91 views

What is the relationship between a brane, a manifold, and a space?

I've read many ways to define manifold; one way is to define it as a type of mathematical space (a type of topological space to be exact). All of the definitions that I've seen for brane, on the ...
2
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0answers
41 views

Quantization of KK Theory

I know that electromagnetism is force via curvature in a U(1)-bundle. I am now trying to literally visualize this, and write down equations that make this manifest. KK (Kaluza-Klein) theory is the ...
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3answers
132 views

Is there any physical interpretation for $\nabla\cdot(\nabla \times F)=0$?

It is well known that the divergence of the curl is always 0. Mathematically I understand why this happens ($d^2=0$ where $d$ is the exterior derivative) but today I was wondering what is the physical ...
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0answers
44 views

Proper time and asymptotic flatness

I'm trying to understand the concept of asymptotic flatness in general relativity, and came up with the following question: If the proper time $\tau$ is infinite for a timelike geodesic, does it mean ...
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0answers
28 views

Is there any reason (other than convenience) to assume the universe is paracompact?

In this discussion on MathOverflow, it is mentioned that the universe, being a Riemannian manifold, must be paracompact. But is there any reason to assume the universe is globally 'small enough'? In ...
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0answers
41 views

Variation of Bazanski Lagrangian

The Bazanski Lagrangian is defined as $$ L=g_{\alpha \beta }U^{\alpha }\frac{D\psi ^{\beta }}{Ds} $$ and $$ U^{\alpha }=\frac{\mathrm{d} x^{\alpha }}{\mathrm{d} s} $$ $x^{\alpha }$ is the ...
19
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1answer
359 views

What, to a physicist, are instantons and the Donaldson invariants?

I study gauge theory from a mathematical perspective. To me, one of the most fundamental ideas is the notion of an instanton on a 4-manifold. To be precise, I have a Riemannian 4-manifold and a ...
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1answer
61 views

Possible inconsistency of mixed index tensor notation

I am posting this here, because in my experience, this sort of thing exists in physics-related works only. Given a local frame $\{e_{(i)}\}$ on some $n$-dimensional manifold $M$, and given a local ...
2
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1answer
47 views

Local Coordinate Expressions for Lie Derivatives

I'm currently working through the math chapters of Norbert Straumann's book on General Relativity. I have trouble understanding the coordinate expression of the Lie derivative of a basis vector. The ...
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1answer
1k views

Equation of a torus

In the recent paper http://arxiv.org/abs/1509.03612, page 37. They say that a torus can be described by the equation $$y^2=x(z-x)(1-x)$$ where $x$ is a coordinate on the base $\mathbb{P}_1$. Could ...
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4answers
160 views

Which tensor describes curvature in 4D spacetime?

I heard these two statements which don't work together (in my mind): In 4D spacetime the curvature is encoded within the Riemann tensor. He holds all the information about curvature in spacetime. ...
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1answer
163 views

Null geodesic equations

If one is constrained to the $xt$ plane, one can define the intersection with that plane of the null hypersurfaces originating at some point $P$ as $$ g_{tt} \frac{d P^t}{d \lambda}\frac{d P^t}{d ...
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1answer
84 views

Variations of actions of (lie algebra valued) differential forms

I have always found it a bit difficult to understand the variation of an action written in differential form language. For example, take the action $$\int tr A\wedge A\wedge A$$ where $A=A_\mu ...
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1answer
55 views

Notation: tetrad indices

I am trying to understand the meaning of upper and lower indices as used in the Newman-Penrose formalism. The tetrad is $\lbrace l^{a},n^{a},m^{a},\overline{m}^{a}\rbrace$, where the upper index ...
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0answers
75 views

Magnetic monopole using differential forms

I'm trying to understand the different variations of the Maxwell's equations using differential forms. The Maxwell's equations are $$dF=0\\ *d*F=J$$ where $F$ is the electromagnetic tensor ($F=dA$) ...
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1answer
95 views

Geodesic equation (free particle)

How to find a coordinate system whose geodesic equation does not have the "Christoffel symbol" term? (i.e. free particle - generalized Newton's second law.)
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0answers
120 views

How the Poisson bracket transform when we change coordinates?

I'm studying the book Geometric Mechanics by Darryl D. Holm and there's one exercise in the book I'm not quite getting what has to be done. The same discussion the author makes in the book is made on ...
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1answer
97 views

Are all maximally symmetric spacetimes constant curvature spacetimes?

A $d$ dimensional maximally symmetric spacetime is a spacetime with the maximum allowed number of Killing vectors. This number is $\frac{d(d+1)}{2}$. Constant curvature spacetimes are spacetimes ...
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0answers
13 views

Global angular forms

In the study of anomaly inflow due to an M5 brane (see for instance the paper by Freed, Harvey, Minasian and Moore), one regularizes the $\delta$ function near a source as $$\delta^{(5)} \rightarrow ...
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0answers
49 views

How to visualize a sphere bundle?

In the paper ``Gravitational Anomaly Cancellation for M Theory Fivebranes", the authors consider removing a tubular region of radius $\epsilon$ around the M5 brane (in order to make sense of the three ...
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1answer
85 views

Time independent Kerr metric

The Kerr metric expressed in terms of polar coordinates $r,\theta,\phi$, such that $x = r\sin(\theta)\cos(\phi)$, $y = r\sin(\theta)\sin(\phi)$, $z = r\cos(\theta)$. Then the Kerr metric is given as ...
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0answers
48 views

Existence of a solution for geodesic differential equations for a singular metric

In order to determine the geodesics, one must solve the following set of differential equations \begin{align} \frac{d^2 x^j}{ds^2} + {j\brace h\,\,k}\frac{dx^h}{ds}\frac{dx^k}{ds} = 0, \end{align} ...
5
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1answer
166 views

Finding diffeomorphism given vector fields [closed]

Given a vector field how do you find the associated diffeomorphisms? Say I am given a vector field in Minkowski space $$\xi = x \frac{\partial}{\partial t} + t \frac{\partial}{\partial x}.$$ How do ...
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1answer
111 views

What is the sum of the angles of a triangle on Earth orbit?

Gauss went out and measured triangles made up of mountain peaks to show that the angles sum up to 180 degree. However, general relativity leads to non-Euclidian space and I would like to get a better ...
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0answers
30 views

Non-time orientable quotient of de Sitter space

Examples of non-time orientable spacetimes are pretty scarce, but it seems the big one is quotients of de Sitter space of the form $dS^n/\pi_1$, where $\pi_1$ is some subgroup of the isometries of de ...
2
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0answers
72 views

Two dimensional spacetime and the Gauss Bonnet theorem

Generally two dimensional spacetimes are deemed to be static, as the Gauss Bonnet theorem implies that the Einstein Hilbert action would be a constant independent of $g$. But as far as I can tell, ...
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2answers
79 views

If curved paths imply that the vehicle is accelerated, how come do we assume that light gets curved whilst its speed is constant?

I don't understand how we can accept these two sentences at the same time: Light speed is constant, therefore experiences no acceleration. On the presence of a gravitation field, light path is ...
3
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5answers
159 views

Local inertial frame

In general relativity we introduce local inertial frames to be such frames where the laws of special relativity holds. Let $\xi^{\alpha}$ the coordinates in the local inertial frame, so we get ...
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0answers
125 views

Cones with deficit angle 2$\pi$ and euler characteristics

I've managed to confuse myself with cones and deficit angles. Let's consider a conical defect in 2 dimensions. So the metric is the usual one in polar coordinates, $$ ds^2 = dr^2 + r^2 d\phi^2,$$ ...
2
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0answers
65 views

Integral curves in null hypersurfaces [closed]

Let be $(M^{n+1},g)$ a spacetime (Lorentz manifold, connexe and time-oriented), $n\ge 2$, and $S\subset M$ a null hypersurface (codim $S=1$ and the restriction of $g$ to each tangent space $T_p S$ ...
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0answers
90 views

What are the implications of integrating the Poisson bracket? [closed]

Reading Ref. 1 I admit I am a little lost in some places. I was hoping that someone in this area could explain the basic premise of integrating the Lie bracket and further is it connected to nlab's ...
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0answers
50 views

How does the expanding of null hypersurface orthogonal geodesic congruence imply a particular result?

Sorry that I do not know how to summarize my problem in the title. First, please go to the website here (free access, even though it looks otherwise) to download the paper done by R. Sashs on ...
4
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0answers
64 views

Trajectories in AdS

On page 2 of this paper (http://arxiv.org/abs/1106.6073), Maldacena explains (and has a very nice picture) showing the trajectories that a timelike and null particle would take in AdS space. Of ...
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1answer
74 views

Operational Definition of Reference Frame in General Relativity

Most treatments of GR begin with the assumption that spacetime is a pseudo-Riemannian manifold (or, sometimes, that it is a more general manifold). But this entails quite a few tacit assumptions about ...
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0answers
32 views

Topics required before studying Tensor Calculus [closed]

Which topics do I need to know before going through tensor calculus?
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1answer
41 views

Expanding the Ricci tensor by summing over indices

I had an attempt at deriving the Schwarzschild metric. This is a 4-dimensional problem where the indices are being summed from 0 to 3. I got up to the part where I calculate the Ricci tensor which is ...
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0answers
33 views

How to derive the relation satisfied by “gravitational magnetic field” from an equation of the Weyl tensor?

Let us call the spacetime $M$ with a metric $g_{ab}$. There is a unit spacelike vector field $\eta^a$ orthogonal to a hypersurface. So that we can define the so-called gravitational electric and ...