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

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3
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1answer
566 views

Derivation of the volume element (which uses the metric tensor)?

I have often seen $\sqrt{-g}$ in integrals, especially actions, where $g=\mathrm{det}(g_{\mu \nu})$. Does anyone know of a derivation that shows that this is indeed the volume element which must be ...
2
votes
1answer
234 views

Parallel transport of a vector along a closed curve in curvilinear coordinates

There is an expression indicating the change of the vector parallel translation along a closed infinitesimal curve in curvilinear coordinates (one way of introducing curvature tensor): $$ \Delta A_{k} ...
3
votes
1answer
281 views

The most general form of the metric for a homogeneous, isotropic and static space-time

What is the most general form of the metric for a homogeneous, isotropic and static space-time? For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) ...
4
votes
1answer
121 views

“WLOG” re Schwarzschild geodesics

Why, when studying geodesics in the Schwarzschild metric, one can WLOG set $$\theta=\frac{\pi}{2}$$ to be equatorial? I assume it is so because when digging around the internet, most references seem ...
1
vote
0answers
170 views

Do we expect that the universe is simply-connected? [duplicate]

I heard recently that the universe is expected to be essentially flat. If this is true, I believe this means (by the 3d Poincare conjecture) that the universe cannot be simply-connected, since the ...
2
votes
2answers
454 views

Ricci tensor for a 3-sphere without Math packets

Let's have the metric for a 3-sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's ...
6
votes
2answers
450 views

First and second fundamental forms

I'm writing notes about the 3+1 formalism in general relativity, for myself. Inevitably I came across the notions of first and second fundamental forms. Mathematically, it is clear how these objects ...
0
votes
1answer
462 views

Contraction of the metric tensor

This is perhaps a simple tensor calculus problem -- but I just can't see why... I have notes (in GR) that contains a proof of the statement In space of constant sectional curvature, $K$ is ...
9
votes
2answers
317 views

Forces as One-Forms and Magnetism

Well, some time ago I've asked here if we should consider representing forces by one-forms. Indeed the idea as, we work with a manifold $M$ and we represent a force by some one-form $F \in ...
2
votes
2answers
337 views

Local inertial coordinates/Fermi normal coordinates

It is said that we can introduce local inertial coordinates/Fermi normal coordinates for any timelike geodesic. But why only for timelike geodesics? What about null geodesics? Perhaps it has to do ...
1
vote
3answers
201 views

Combining metric tensors/curvature tensors

I was thinking about the following scenario: Consider a particle which causes a metric $g_{\mu\nu}$ on an otherwise Minkowski spacetime (or any manifold). Now, consider another particle, somewhere in ...
2
votes
0answers
165 views

Why doesn't this metric cover all of de Sitter space?

This represents a confused attempt to work through a problem in Carroll's Spacetime and Geometry. Supposedly I should be able to use the geodesic equation, ...
-1
votes
2answers
1k views

What is metric of spherical coordinates $(t,r,\theta,\phi)$?

In spherical coordinates the flat space-time metric takes: $$ds^2=-c^2dt^2+dr^2+r^2d\Omega^2$$ where $r^2d\Omega^2$ come from when the signature of metric $g_{\mu\nu}$ is (-,+,+,+)? what is ...
3
votes
1answer
111 views

Energy Functional

I am a graduate student in pure mathematics, during my study on Ricci Flow I faced some functional known as energy functional. For example Einstein-Hilbert functional is called an energy functional, ...
1
vote
2answers
157 views

What is path of light in the accelerating elevator?

Mathematically, (by mathematically I means by equations) what is path of light in the accelerating elevator? What is the difference between an ordinary derivative and covariant derivative (which is ...
1
vote
0answers
72 views

Geometry for Physics [duplicate]

I am currently a high school student interested in a research career in physics. I have self taught myself single variable calculus and elementary physics upto the level of IPHO . And I am comfortable ...
7
votes
0answers
279 views

Extended Born relativity, Nambu 3-form and ternary (n-ary) symmetry

Background: Classical Mechanics is based on the Poincare-Cartan two-form $$\omega_2=dx\wedge dp$$ where $p=\dot{x}$. Quantum mechanics is secretly a subtle modification of this. By the other hand, ...
2
votes
1answer
476 views

Cartan equations versus Einstein equations in classical gravity

Are Cartan structural equations equivalent to Einstein's equations $$G_{\mu\nu}=T_{\mu\nu}$$ and why (in the case of torsionless geometries, of course)? Does it also apply with a non-null ...
2
votes
1answer
547 views

Ricci identity/Riemann curvature tensor and covectors

Can somebody please explain to me how the following statement is true? The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity $$(\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ...
3
votes
2answers
159 views

Electric Field One-Form

I know for instance that we can interpret the electric field as the one-form that given a vector gives the change in potential in the direction of the vector, however I'm very unsure about how to ...
1
vote
0answers
145 views

Why are differential forms on a n-dimensional manifold a Grassmann algebra?

This is stated as an obvious example of a Grassmann algebra on page 32 in this tutorial I am trying to read, but to me it is unfortunately not so obvious. So can somebody expand this comment a bit ...
3
votes
1answer
622 views

Change of coordinates from an arbitrary frame to a locally inertial frame in General Relativity

If I have the following metric: $$ds^2=(1-2\phi)c^2 dt^2 - (1-2 \phi)(dx^2+dy^2+dz^2)$$ $\phi$ being the gravitational potential with $|\phi| << 1$ everywhere. How do I find a coordinate ...
0
votes
0answers
50 views

Curved space to flat space calculation

When changing the curved space co-ordinate into a flat space co-ordinate if a cone. I got the result transformation that i cannot get a transformation at the vertex(apex) why?
11
votes
3answers
971 views

What is a dual / cotangent space?

Dual spaces are home to bras in quantum mechanics; cotangent spaces are home to linear maps in the tensor formalism of general relativity. After taking courses in these two subjects, I've still never ...
8
votes
2answers
1k views

Dirac equation in curved space-time

I have seen the Dirac equation in curved space-time written as $$[i\bar{\gamma}^{\mu}\frac{\partial}{\partial x^{\mu}}-i\bar{\gamma}^{\mu}\Gamma_{\mu}-m]\psi=0 $$ This ...
1
vote
1answer
139 views

Where to read about Minkowski space [duplicate]

When I learned Special Relativity, it was taught in terms of basic linear algebra, without any mention of the Minkowski space, proper time as integration on the metric, etc. However, when I am trying ...
5
votes
1answer
596 views

Fourier Transform on a Riemannian Manifold

The question is quite simple: What would be the definition of Fourier Transform (and it's inverse) on a Riemannian Manifold? I've found that a similar question has been asked at Mathematics.SE but ...
3
votes
2answers
229 views

Differential Forms and Densities

I've heard that differential forms are related to densities, however I'm still a little confused about that. I thought on the case of charge density and I came to that: let $U\subset\mathbb{R}^3$ be a ...
2
votes
1answer
148 views

Newtonian Gravity on a Riemannian $3$-Manifold

To solve the Poisson equation for the Newton Potential, say $\phi$, one can use the divergence theorem, such that $$\int_U \nabla^2 \phi \sqrt{g}~ dV= \int_{\partial U} <\nabla \phi,n> ...
22
votes
4answers
2k views

Physical and Geometrical interpretation of Differential Forms

I have a doubt about the physical and geometrical interpretation of differential forms. I've been studying differential forms on Spivak's Calculus on Manifolds, but my real intent is to use those ...
6
votes
2answers
215 views

Are Poisson brackets of second-class constraints independent of the canonical coordinates?

Say we have a constraint system with second-class constraints $\chi_N(q,p)=0$. To define Dirac brackets we need the Poisson brackets of these constraints: $C_{NM}=\{\chi_N(q,p),\chi_M(q,p)\}_P$ . Is ...
3
votes
3answers
440 views

Polyakov action: difference induced metric and dynamical metric

The Polyakov action is given by: $$ S_p ~=~ -\frac{T}{2}\int d^2\sigma \sqrt{-g}g^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu} ~=~ -\frac{T}{2}\int d^2\sigma ...
15
votes
3answers
868 views

Representing forces as one-forms

First of all, sorry if any of those things are silly or nonsense, I'm just trying to understand better how the concepts of forms, exterior derivative and so on can be used in physics. This question ...
0
votes
1answer
70 views

Constraint on a metric

Given a metric of the form $$ds^2=dr^2+a^2\tanh^2(r/b)d\theta^2$$ why does it follow that $a=b$? I can't quite spot a constraint condition...
0
votes
1answer
43 views

What is the Willmore energy of the Earth (or the geoid)?

Wikipedia defines the Willmore energy as: $$e[{\mathcal{M}}]=\frac{1}{2} \int_{\mathcal{M}} H^2\, \mathrm{d}A,$$ where $H$ stands for the mean curvature of the manifold $\mathcal{M}$. What is the ...
2
votes
1answer
759 views

Covariant derivative

I would very much appreciate some help in The following: What is 2nd order covariant derivative $$\nabla_i\nabla_jf(r)$$ in terms of $r,\theta, g(r)$ and partial derivative, given that the metric ...
3
votes
2answers
1k views

Null geodesic given metric

I (desperately) need help with the following: What is the null geodesic for the space time $$ds^2=-x^2 dt^2 +dx^2?$$ I don't know how to transform a metric into a geodesic...! There is no need to ...
4
votes
1answer
144 views

Flat space metrics

This question concerns the metric of a flat space: $$ds^2=dr^2+cr^2\,\,d\theta^2$$ where $c$ is a constant. Why is it necessary to set $c=1$ to avoid singularities and to restrict $r\ge 0$? Thanks.
1
vote
1answer
298 views

Interpreting Vector fields as Derivations on Physics

I have a subtle doubt about the physical interpretation of the mathematical definition of vector field as a derivation. In basic physics we understand a vector quantity as a quantity that needs more ...
9
votes
2answers
472 views

Does Kaluza-Klein Theory Require an Additional Scalar Field?

I've seen the Kaluza-Klein metric presented in two different ways., cf. Refs. 1 and 2. In one, there is a constant as well as an additional scalar field introduced: $$\tilde{g}_{AB}=\begin{pmatrix} ...
0
votes
0answers
80 views

Is there a book that discusses General Relativity in terms of Modern Differential Geometry? [duplicate]

All of the physics books that I've seen which discuss General Relativity do so in terms of coordinates - the tensor calculus - even though the naturally relevant entities are invariant under general ...
3
votes
2answers
634 views

Metric coefficients in rotating coordinates

Let $(t,x,y,z)$ be the standard coordinates on $\mathbb{R}^4$ and consider the Minkowski metric $$ds^2 = -dt^2+dx^2+dy^2+dz^2.$$ I am trying to compute the metric coefficients under the change of ...
5
votes
1answer
224 views

Kaluza-Klein Christoffel Symbols

I have a question regarding the connection coefficients as they pertain to the following paper: http://www.weylmann.com/kaluza.pdf . When I try to calculate the 4D Christoffel symbols from the 4D part ...
1
vote
2answers
395 views

What is the Riemann curvature tensor contracted with the metric tensor?

Can the Ricci curvature tensor be obtained by a 'double contraction' of the Riemann curvature tensor? For example $R_{\mu\nu}=g^{\sigma\rho}R_{\sigma\mu\rho\nu}$.
5
votes
1answer
260 views

Do partial derivatives commute on tensors?

Do partial derivatives commute on tensors? For example, is $$\partial_{\rho}\partial_{\sigma}h_{\mu\nu} - \partial_{\sigma}\partial_{\rho}h_{\mu\nu}=0$$ correct?
5
votes
1answer
111 views

Help with the understanding of boundary conditions on $AdS_3$

So I am trying to reproduce results in this article, precisely the 3rd chapter 'Virasoro algebra for AdS$_3$'. I have the metric in this form: ...
6
votes
1answer
2k views

Physical significance of Killing vector field along geodesic

Let us denote by $X^i=(1,\vec 0)$ the Killing vector field and by $u^i(s)$ a tangent vector field of a geodesic, where $s$ is some affine parameter. What physical significance do the scalar quantity ...
3
votes
1answer
431 views

Can one raise indices on covariant derivative and products thereof?

Can the following be true? $g^{\sigma\rho}\nabla_{\rho}\nabla_{\mu} = \nabla^{\sigma}\nabla_{\mu}$ $g^{\sigma\rho}\nabla_{\nu}\nabla_{\sigma} = \nabla_{\nu}\nabla^{\rho}$ ...
1
vote
1answer
330 views

Covariant derivative with upper index

I just need clarification, that is, to see that I'm doing the right thing. When calculating central charge for certain metric, I need to solve an integral that contains Lie brackets etc. And I have ...
2
votes
1answer
608 views

Difference between $\partial$ and $\nabla$ in general relativity

I read a lot in Road to Reality, so I think I might use some general relativity terms where I should only special ones. In our lectures we just had $\partial_\mu$ which would have the plain partial ...