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

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2
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1answer
73 views

Rindler and Minkowski space future/past infinity

In my black holes course, we are looking at the Penrose diagram for 1+1 D Minkowski space. My notes don't specifically describe $i^{\pm}$ (future/past timelike infinity) but do say all timelike curves ...
1
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0answers
37 views

SR: vector field and change of reference [closed]

If $U$ and $V$ are vector fields, then the derivative of $U$ along $V$ is the vector field $\nabla _V U$ with components $$\nabla _V U^a=V^b \frac{\partial U^a}{\partial x^b}.$$ I would like to verify ...
1
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1answer
75 views

Can the N-body problem be solved numerically using the geodesic equation of mass-distorted spacetime?

I would like to write a program that solves the trajectories of objects (think rockets) that are influenced by mass of other objects (think planets). I saw that I can do this using Newton's laws, but ...
0
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0answers
52 views

Connection one-form and suppressed indices

I am reading Sean Carroll's notes on GR, which states (Page 91): Using our freedom to suppress indices on differential forms, we can write the defining relations for these two tensors as: $$ T^a ...
2
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1answer
90 views

Lagrangian vector field expression

The Lagrangian vector field $X_L$ on the tangent bundle is given in page 4 of Marco Mazzucchelli's "critical Point Theory for Lagrangian systems" to be; \begin{equation} ...
1
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2answers
79 views

Allowable spacetime deformations [closed]

What deformations are possible with spacetime? By 'deformation' I am referring to the kind of change in spacetime caused by the presence of a mass which deforms spacetime sufficiently to deflect ...
1
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0answers
28 views

Singular points of an orbit space

I am wondering what, precisely, the singular point of an orbit space is. Specifically, I am looking at quantum statistics and the orbit space $M^N/S_N,$ where $M^N$ is the classical configuration ...
0
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3answers
136 views

Definition of non-degenerate metric tensor

We know that a metric has a property which is called non-degeneracy. I was searching for what does that mean and saw it associated with the fact that $det(g_{\mu\nu})\neq0$. How does this relate to ...
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2answers
81 views

Can someone explain how Weinberg's definition of the affine connection for the geodesic equation matches the definition of an affine connection?

Consider the geodesic equation \begin{equation} 0=\frac{d^2 x^\lambda}{d\tau^2}+ \Gamma^\lambda_{\mu\nu} \frac{d x^\nu}{d\tau}\frac{d x^\mu}{d\tau} \end{equation} In Gravitation and Cosmology, on page ...
2
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4answers
228 views

Normal Vectors to these Hypersurfaces on a Lorentzian Manifold

With respect to the coordinates $(x^{0},x^{1},x^{2},x^{3})=(v,r,\theta,\phi)$, we have the following components of the metric tensor: $\begin{bmatrix} g_{00} & g_{01} & g_{02} & ...
1
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1answer
70 views

Bianchi Identity using null tetrad

I'm currently looking at the Newman-Penrose Formalism, and trying to understand where there sets of equations come from. For that, I need to know how I can write the second Bianchi identity for the ...
-1
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1answer
59 views

Weyl scalar calculation

I'm trying to compute Weyl scalars, but don't really understand the formulae for them, in the sense I don't understand how to compute them. Let's take ...
1
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4answers
289 views

Kerr metric Christoffel symbols

I've been slaving away trying to calculate the Christoffel symbols for the Kerr metric. Does anybody know of a link that I could compare my answers to? I've done some Google searches and all I can ...
5
votes
3answers
189 views

How can a Physical law not be invariant?

In Relativity, both the old Galilean theory or Einstein's Special Relativity, one of the most important things is the discussion of whether or not physical laws are invariant. Einstein's theory then ...
3
votes
0answers
70 views

Feynman Path integrals in space with holes in it [closed]

Feynman Path Integrals are a way of calculating the wave function of quantum mechanics. It usually integrates every possible path through all of space. I wonder if there is any study of Feynman path ...
11
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3answers
1k views

How many times can light revolve around a black hole?

Take a light ray approaching a black hole from infinity which goes out again to infinity. What is the maximum finite rotation it can describe? (I know it can loop around indefinitely in the ...
2
votes
0answers
156 views

Covariant versus “ordinary” divergence theorem

Let $M$ be an oriented $m$-dimensional manifold with boundary. As stated in Harvey Reall's general relativity notes (here) or Sean Carroll's book, the "covariant" divergence theorem (i.e. with ...
1
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2answers
243 views

Is the universe 5 dimensional space-time or 4?

we've been told that in General Relativity (GR), matter tells space how to curve and space tells matter how to move. But my question is, if 3 dimensional space was curved by matter then it should be ...
4
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0answers
59 views

How many Killing spinors exist on $S^5$?

So, I know that on $S^n$, a spinor of the form $$ \Sigma^\pm = \frac{1 \pm i\gamma^\alpha z_\alpha}{\sqrt{1+z^2}}\Sigma_0$$ where $\Sigma_0$ is a constant spinor, is a Killing spinor on $S^n$ ...
1
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1answer
134 views

Distributions (generalized functions) over manifolds

I have asked a similar question on the math stackexchange website, but since this type of question might have an answer that is known to physicists better than mathematicians I'm posting the question ...
11
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7answers
289 views

How can a set of components fail to make up a vector?

Many books in Physics insist to define vectors are objects with components with the property that the components transform in a proper way under a change of coordinates. Now, in mathematics, on the ...
1
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1answer
84 views

Is my Summary of a Spinor Bundle Associated with a String Worldsheet Correct?

I've been having difficulty finding a source that lists all the properties of the spinor bundle of a string worldsheet explicitly, so I've had a go at creating my own description. I'd really ...
2
votes
5answers
234 views

A reference frame is any coordinate system or just a set of Cartesian axes?

In Physics the idea of a reference frame is one important idea. In many texts I've seem, a reference frame is not defined explicitly, but rather there seems to be one implicit definition that a ...
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0answers
26 views

Free Components of the Riemann Tensor

Knowing the symmetries of the Riemann tensor, it is known that in 4-dimensional space we would have only 20 free components. My question is: How one can decide which components are necessary to ...
0
votes
1answer
66 views

Lie derivative in this paper [closed]

In this paper http://arxiv.org/abs/1210.2332 it says in (3.19) p. 8 that $$L_{V}z^A =0$$ but I don't know much about Lie derivatives except what I saw now through wikipedia ...
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0answers
24 views

2D CFT for nontrivial topology

What is a systematic way to calculate a general $N$-points correlation function of 2D CFT for a nontrivial topology? Piece by piece of this can be found in many different CFT and String Theory ...
0
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1answer
63 views

Will a stress-energy tensor have the same identities as it's metric?

Say I have a metric tensor where $$g_{00} = -c^{2}\ and $$ $$g_{01}=g_{02}=g_{03}=0$$ and $$g_{12}=g_{13}=g_{23}$$ and $$g_{11}=g_{22}=g_{33}$$ My question is straightforward: would the same or ...
5
votes
1answer
240 views

Does magnetic monopole violate $U(1)$ gauge symmetry?

Does a magnetic monopole violate $U(1)$ gauge symmetry? In what sense and why? Insofar as I know, there are at least two types of magnetic monopoles. One is the Dirac monopole while the other is the ...
0
votes
1answer
84 views

Geodesic equation

I have a technical question about the geodesic equation. Assume we have a frame $(E_{1},E_{2},E_{3},E_{4})$ (not necessarily a coordinate frame). Assume we have a parametrized curve $\gamma(s)\in M$ ...
0
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3answers
362 views

Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?

I am dissatisfied with the presentation (not to say "definition") of "arc length", in its "Generalization to (pseudo-)Riemannian manifolds", as given in Wikipedia. (Who isn't?. But I'll sketch it here ...
1
vote
1answer
95 views

Understanding the covariant derivative and its relation to parallel transport

I have been reading section 3.1 of Wald's GR book in which he introduces the notion of a covariant derivative. As I understand, this is introduced as the (partial) derivative operators $\partial_{a}$ ...
0
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0answers
64 views

The Covariant Spinor Derivative in the Locally Supersymmetric Generalisation of the Polyakov Action and Potential Mistakes in the Literature

Questions) I recently came upon the thesis The Landscape of Free Fermionic Gauge Models by D. G. Moore and G.B. Cleaver. On pages 20 and 21 they explain that the locally supersymmetric action, ...
2
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2answers
87 views

Symplectic structure and isomorphisms

In his book Mathematical Methods of Classical Mechanics, V.I. Arnold writes To each vector $\xi$, tangent to a symplectic manifold $(M^{2n},\omega^2)$ at the point $\mathbf{x}$, we associate a ...
2
votes
1answer
176 views

Inverse Metric Tensor

First the setup: Let $\mathcal M$ be a $2$-dimensional manifold. Let $U_P$ be some open neighbourhood of a point $P \in \mathcal M$. Let $\mathcal F : U_P \rightarrow \mathbb R \times \mathbb R$ be ...
2
votes
1answer
93 views

Fastest way to find the curvature terms from a given metric [closed]

I want to find the spherically symmetric, static solutions to Einstein's equations $$ R_{\mu \nu} - \frac{1}{2}Rg_{\mu \nu} = 0 $$ in four dimensions using the metric $$ g_{\mu \nu}dx^{\mu}dx^{\nu} ...
0
votes
1answer
41 views

Given any metric, how to find the straight line path between two points? [closed]

Say we are given a two-dimensional metric $$ds^2=f_1(x)dx^2+f_2(x)dy^2,$$ for any kind of function. How do we calculate the distance along a straight line path (not the shortest possibly) between, ...
1
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0answers
120 views

Quadratic order perturbation terms in the expansion of Ricci tensor [closed]

I want to expand Einstein-Hilbert action for the metric $$ g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} $$ up to quadratic order in $h_{\mu \nu}$. For this purpose I need to calculate the Ricci tensor ...
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2answers
185 views

Is there an accepted axiomatic approach to general relativity?

I am reading Steven Weinberg's book Gravitation and Cosmology. He makes a big deal out of the equivalence principle and showed a bunch of deductions you can make based on it. This surprised me since ...
2
votes
1answer
97 views

Doubt regarding the dimension of a manifold

I'm very unsure about all this business so please forgive any inaccuracies in my question. Essentially what I'm having trouble understanding right now is how we "decide" the dimensionality of a ...
0
votes
1answer
51 views

Invariance of the low energy effective string action

It is well known that the action of General Relativity $$S = \frac{1}{16\pi G}\int R\;\sqrt{-g} d^D X$$ is invariant under "diffeomorphisms". The low energy effective action for bosonic strings is ...
3
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0answers
104 views

Chocolate dynamics

Now I have found a possible model on how to describe chocolate when it is chewed. It has to do with geometrical transformations when a curve $\gamma$ intersects a manifold $M$. The chocolate is ...
1
vote
1answer
43 views

What does “local composite symmetry” mean in ${\cal N}=8$ $d=5$ supergravity?

What does it mean "local composite symmetry" in supergravity? Specifically, I don't understand very well the local composite symmetry ${\rm USp}(8)$ in ${\cal N}=8$ $d=5$ supergravity.
6
votes
2answers
221 views

Is a spinor in some sense connected to space?

Spinors transform under the representation of $SL(2,\mathbb{C})$ which is the double cover of the Lorentz group $SO(1,3)$ - or in the non-relativistic case under $SU(2)$, the double cover of $SO(3)$. ...
0
votes
0answers
76 views

Penrose diagram (Reissner-Nordstrom metric)

I try to derive the Penrose diagram for the Reissner-Nordstrom metric $$ \text d s^2 = -\frac{(r-r_+)(r-r_-)}{r^2}\text d t^2 + \frac{r^2}{(r-r_+)(r-r_-)}\text d r^2 + r^2 \text d \Omega^2\;,\qquad ...
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0answers
33 views

Assigning an asymptotic power to the volume form?

I was reading about the covariant theory of asymptotic symmetries in this review: http://arxiv.org/abs/hep-th/0111246 I have a question about eq. (1.8), but before I ask I should describe what the ...
2
votes
1answer
176 views

Are the Jacobi equation and the geodesic deviation equation related?

On page 111 in his book Riemannian Geometry, Manfredo Do Carmo states what he calls the Jacobi equation \begin{equation} \frac{D^2J}{dt^2} + R(\gamma'(t),J(t))\gamma'(t) = 0 \end{equation} ...
2
votes
1answer
135 views

Spin connection and covariant derivative

How to prove explicitly (i.e., to don't postulate it) that by including Lorentz indices $a$ the covariant derivative $D_{\mu}$ looks like $$ D_{\mu}A^{\nu a} = \partial_{\mu}A^{\nu a} + ...
0
votes
1answer
80 views

What does this Hodge dual symbol $\star_3$ mean?

We know that in this $$\star {f(...)}$$ the $\star$ represents the Hodge dual. But in this: $\star_3 f(...)$ what does specifically the $\star_3$ symbol mean?
0
votes
2answers
148 views

Properties of Hodge Duality

So we know that Hodge duality works this way $$⋆(dx^i_1 \wedge ... \wedge dx^i_p)= \frac{1}{(n-p)!} \epsilon^{i_1..i_p}_{i_{p+1}..i_n} dx^{i_{p+1} } \wedge dx^{i_n}$$ where $p$ represents the $p$ in ...
10
votes
4answers
706 views

Can a metric in General Relativity, Supergravity, String Theory, etc., be asymmetric?

Why is it that all problems I encountered until now have metrics that when represented in a matrix form turn out to be symmetric? Aren't there asymmetric matrices representing some metrics?