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

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51 views

Spinor notation in general relativity

I have a somewhat broad/big question, and I know that there are many references for it available out there. However, so far I couldn't find anything that I can really understand, that's why here is my ...
2
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1answer
32 views

Parallel Transported Orthonormal Basis

The following argument results in a conclusion that I find strange, and makes me suspect there is something wrong with the reasoning. First, consider a timelike geodesic $\gamma$ with normalized ...
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2answers
49 views

Deriving the geodesic equation [on hold]

I having been reading a general relativity book, but when in comes to the geodesic equation, it is not derived. How does one go about doing this.
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0answers
45 views

metric determinant and its partial and covariant derivative

question : $\nabla_a \nabla_b \sqrt{g} \phi =\partial_a \sqrt{g} \partial_b \phi$ is true ? because $\nabla_a \sqrt{g}=0$ so we can write $\sqrt{g} \nabla_a \nabla_b \phi$ , but because metric ...
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1answer
63 views

Varying wrt metric [on hold]

I saw people write $\frac{\partial( F^{ab} F_{ab})}{\partial g^{ef}}$ as $\frac {\partial (g^{ca}g^{db}F_{cd}F_{ab})}{\partial g^{ef}}$ in a way that exposes the dependence on the metric. but ...
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0answers
25 views

Books or papers recommendation on orbifold and CFT

Could you recommend some references on orbifold CFT? I have found this paper "The conformal field theory of orbifolds"(1987)(http://inspirehep.net/record/230342) is very useful for me, so I want to ...
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1answer
27 views

Why does this allegedly Hermitian Kähler metric have non-zero diagonal terms?

In string theory, the Kähler potential of Kähler moduli (e.g. - the volume of a Calabi-Yau manifold) is given by (see, for instance, Becker, Becker, Schwarz: "String Theory and M Theory" p. 498) $$K ...
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1answer
44 views

Show that a solenoidal field is always a curl of a vector field [closed]

Can someone prove that: $$\nabla \cdot \mathbf{B} = 0 \implies \mathbf{B} = \nabla \times \mathbf{A}~?$$ I know that $$\nabla \cdot (\nabla \times \mathbf{A}) = 0$$ identically. But can one prove ...
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0answers
33 views

Killing vector field along geodesic [closed]

I was trying to show that a Killing vector field satisfies the Jacobi Equation for a geodesic, just by assuming that \begin{equation} \nabla_\mu X_\nu + \nabla_\nu X_\mu=0 \end{equation} Indeed, if I ...
5
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1answer
94 views

Momentum is a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
8
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1answer
99 views

What is the physical interpretation of the Poisson bracket [duplicate]

Apologies if this is a really basic question, but what is the physical interpretation of the Poisson bracket in classical mechanics? In particular, how should one interpret the relation between the ...
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0answers
36 views

Jets and vertical differential [migrated]

For a vector bundle $(E,\pi, M)$ let $\phi :M\mapsto E$ be a section of $\pi $, $x\in M$ and $u=\phi (x)$. The vertical differential of the section $\phi$ at point $u\in E$ is the map: ...
2
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2answers
50 views

Under what representation do the Christoffel symbols transform?

I often read the statement, that the Christoffel symbols aren't tensors. But then, under which representation do they transform?
2
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2answers
72 views

Differential operators in curvilinear coordinates

In the appendix A of Griffith's Electrodynamics text, he cites Spivak's Calculus on Manifolds as a reference more a more complete treatment of taking the gradient, curl, divergence, and Laplacian in ...
1
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1answer
90 views

Covariant derivative of Levi-Civita tensor [closed]

I'm currently studying Carroll's GR book Spacetime & Geometry, and ran into some trouble understanding the text at page 99 which says: By using metric compatibility $$(\nabla_{\alpha}g)_{\mu ...
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2answers
82 views

Phase space Lagrangian?

Reading out of this lecture series we define a phase space Lagrangian $\mathcal L$ to be a function of $4n+1$ variables namely $q,\dot q,p,\dot p,t$. My question is, what space is this function ...
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0answers
140 views

Can the universe be round but still infinite?

Can the universe still be infinite in space if its curvature is > 1? Is a manifold of positive curvature necessarily compact? Does the Tarski paradox have any bearing on the finite or infinite ...
8
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1answer
121 views

What does Ricci tensor do with two vectors?

I have found it easier to understand the meaning of a particular tensor if I can find out what does it do if I cancel all its lower indices with vectors in short: $g_{ij} u^i v^j$: dot product of ...
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0answers
49 views

Is the Weyl Postulate correct?

The Weyl postulate in cosmology states that worldlines do not intersect but it can be shown in GR that using Raychaudhuri equation that geodesics can intersect if there is curvature so I'm really ...
2
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1answer
44 views

Local translations in curved spacetime

A global Poincare transformation on a scalar field induces $$\delta(a, \lambda)\phi(x) = [a^{\mu}+\lambda^{\mu\nu}x_{\nu}]\partial_{\mu}\phi(x). \tag{11.46}$$ In curved spacetime we replace $a^{\mu} ...
1
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1answer
129 views

Prove Christoffel Symbol Identity

In a book I am reading, the following identity is claimed and then "left to the reader to prove." $g_{ij}$ is the metric tensor, and $\Gamma$ is the Christoff symbol of the second kind with the ...
3
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0answers
66 views

Maxwell's equation in curved spacetime - how come? And experimental evidence?

I'm trying to understand the generalization of Maxwell's equations to curved spacetime. In FLAT (Minkowski) SPACETIME: If we define the "four-potential" as $$\ ...
2
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1answer
56 views

When can an autonomous system be written using a Hamiltonian?

If I have an autonomous series of differential equations $$\tag{1} \frac{dx_i}{dt} ~=~ A_i(x_1,...,x_n)$$ with the condition that $$\tag{2} \sum_{i=1}^n\frac{\partial A_i}{\partial x_i}~=~0$$ in all ...
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0answers
47 views

How is Infinitesimal coordinate transformation related to Lie derivatives?

I am reading the book "Gravitaion and Cosmology" by S. Weinberg. In section 10.9, while discussing Lie derivatives of tensors of different ranks, he makes a general comment: The effect of an ...
3
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1answer
55 views

In continuum mechanics, why is the stress vector $T=\sigma\cdot n$ not a covector?

In continuum mechanics, the stress vector (see Cauchy stress tensor) $T=\sigma\cdot n$ is the surface density of a force. Forces are covectors, since they map a displacement vector to a scalar energy. ...
0
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1answer
57 views

EM Field tensor of a point charge [closed]

If I say the Reissner-Nordstrom metric $$ ds^2=-\left(1-\frac{2m}{r}+\frac{e^2}{r^2}\right)\text d t^2 + \left( 1-\frac{2m}{r}+\frac{e^2}{r^2}\right)^{-1}\text d r^2 + r^2 \text d \Omega^2 $$ is the ...
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0answers
17 views

Atlas 2 For Mathematica. Need to calculate Riemann tensor, etc [migrated]

Is anyone familiar with Atlas 2 for Mathematica to calculate the Riemann Tensor, Ricci Tensor, and scalar I have a metric that I need to calculate these things for. Can anyone help? I'm not too up on ...
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0answers
84 views

Integrability in classical mechanics

An integrable system in classical mechanics is defined by action-angle variables and closed loop trajectories in phase space. I have also heard that the flow lines of an integrable system are ...
3
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1answer
63 views

What is the physical cause that circulation on a closed surface is zero?

This is quoted from Feynman's Lectures: We would like to see what happens when the loop shrinks down to a point, so that surface boundary disappears - the surface becomes closed. Now, if the ...
1
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1answer
60 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. ...
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2answers
55 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 ...
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0answers
39 views

$U(1)$ connection and spacetime basis $e^{\mu}$

When dealing with supergravity, it is said that a Kahler-Hodge manifold has a $U(1)$ bundle whose first Chern class coincides with the Kahler class, thus locally the $U(1)$ connection can take the ...
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0answers
34 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 ...
0
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1answer
77 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 ...
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2answers
146 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
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1answer
50 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 ...
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0answers
25 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
63 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
45 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 ...
3
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1answer
80 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} ...
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0answers
46 views

Topic Suggestions - Kaehler-Einstein metrics in physics [closed]

I am hoping to get some topic suggestions for a presentation I have to give in a couple of weeks. The course the presentation is for is called Kahler-Einstein metrics. I would really like the ...
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2answers
72 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
24 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
60 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 ...
2
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2answers
53 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
134 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
54 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 ...
0
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1answer
44 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 ...
2
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4answers
183 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
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3answers
142 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 ...