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

learn more… | top users | synonyms (1)

1
vote
2answers
166 views

Question about basic formalism of GR and the metric tensor

I really don't know much about GR, but I've come across a few rough sketches of its formalism in my DG books. I'm trying to piece it together to get a very basic intuition of what spacetime is in GR. ...
1
vote
2answers
144 views

Metric and the Lagrangian [duplicate]

Does the Lagrangian formalism require a metric on the configuration manifold $Q$ in order to define a Lagrangian $L$ on the tangent bundle $TQ$? Further, if we specify a metric on the tangent bundle ...
1
vote
0answers
69 views

Ricci tensor of Metric black holes with nils and solv geometries of Horizon

The metric of black holes with nils and solv geometries of the horizon is generically represented by $$\mathrm{d}s^2=-r^{2z}\mathrm{d}t^2+\frac{\mathrm{d}r^2}{r^2}+\sum_{I=1}^3 r^{2q_I}(w^I)^2$$ How ...
1
vote
1answer
344 views

What are the mathematical models for force, acceleration and velocity?

In mechanics, the space can be described as a Riemann manifold. Forces, then, can be defined as vector fields of this manifold. Accelerations are linear functions of forces, so they are covector ...
1
vote
2answers
279 views

$\nabla^{\mu}\nabla_{\mu}$ in general relativity [closed]

I am trying to work out $\square=\nabla^{\mu}\nabla_{\mu}$ in the metric $ ds^{2}=-A(r)dt^{2}+B(r)^{-1}dr^{2}+r^{2}d\Omega^{2} $$ My work: when applying $\square$ to a scalar $\phi$, then $ \...
2
votes
1answer
113 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 ...
3
votes
1answer
111 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
votes
0answers
59 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
votes
1answer
104 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} X_L=\sum^M_{j=1}\bigg(v^j\frac{...
1
vote
2answers
94 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
vote
0answers
32 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
votes
3answers
356 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 ...
1
vote
2answers
119 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
votes
4answers
432 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} & g_{...
1
vote
1answer
136 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
votes
1answer
106 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 $\Psi_{2}=C_{1342}=C_{pqrs}l^{p}m^{q}\bar{m}^{r}...
1
vote
4answers
613 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
263 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
84 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
votes
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
405 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
vote
2answers
489 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
votes
0answers
68 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
vote
1answer
153 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
votes
7answers
338 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
vote
1answer
96 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
797 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 ...
1
vote
0answers
36 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 ...
1
vote
1answer
105 views

Lie derivative in this paper [closed]

Say, $$L_{V}z^A =0$$ but I don't know much about Lie derivatives except what I saw now through wikipedia http://en.wikipedia.org/wiki/Lie_bracket_of_vector_fields#Definitions that it is (if I am ...
0
votes
1answer
67 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
434 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
101 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
votes
3answers
380 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
159 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 $\partial_{a}$ ...
2
votes
2answers
116 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 1-...
2
votes
1answer
515 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
149 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
51 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, say,...
1
vote
2answers
348 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
105 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
55 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 $...
4
votes
0answers
129 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
58 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.
8
votes
2answers
271 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 $SO(3)$. ...
1
vote
0answers
37 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
385 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} ...
1
vote
1answer
259 views

Spin connection and covariant derivative [closed]

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} + \Gamma^{\nu}_{\...
1
vote
1answer
100 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
200 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
989 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?