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

learn more… | top users | synonyms

2
votes
2answers
36 views

Geodesic Equation from variation: Is the squared lagrangian equivalent?

It is well known that geodesics on some manifold $M$, covered by some coordinates ${x_\mu}$, say with a Riemannian metric can be obtained by an action principle . Let $C$ be curve $\mathbb{R} \to M$, ...
1
vote
1answer
53 views

Why do derivatives act on vector fields on a worldsheet?

The covariant derivative of a vector $A^{\mu}$ at a point $x$ is defined as $$D_z A^{\mu}=\partial_zA^{\mu}+\Gamma^{\mu}_{\rho\sigma}(x)\partial_{z}x^{\rho}A^{\sigma}$$ where Greek symbols are ...
1
vote
3answers
63 views

All geodesics are inextendable?

I think the title is true, because geodesics has a tangent vector with a constant length parametrized by an affine parameter. Probably, it is easier to think about timelike or spacelike geodesics. ...
3
votes
2answers
309 views

Why don't global coordinates always exist for a manifold?

Let $M$ be a manifold and $(\phi,U)$ a patch. Then $\phi(P)=\bar{x}=\begin{bmatrix} x^1\\ x^2\\ \vdots\\ x^n \end{bmatrix}$ for each $P$ in $U$. But each $P$ in $M$ is in some patch, so this ...
0
votes
0answers
53 views

Covariant Derivative Chain rule?

I want to prove that a covariant derivative of a vector $A^{\mu}(x(z))$ at the point $x(z)$ in general would be defined as $$D_z ...
1
vote
2answers
54 views

Proper time in general relativity

For general relativity, Wald's GR states that timelike curves, with the norm $g_{ab}T^{a}T^{b} < 0$, can be parameterized by the "proper time" $$\tau = \int (-g_{ab}T^{a}T^{b})^{1/2} dt.$$ This ...
0
votes
0answers
52 views

Writing this in a neater more compact way [on hold]

If one has $$ (\frac{d\omega^2}{dx^1} - \frac{d\omega^1}{dx^2}) (\theta^1 \wedge \theta ^2) +(\frac{d\omega^3}{dx^2} - \frac{d\omega^2}{dx^3}) (\theta^2 \wedge \theta ^3) +(\frac{d\omega^1}{dx^3} - ...
1
vote
0answers
60 views

(Scalar) Ricci flatness of a metric

What is the physical meaning to vanishing Ricci scalar $R=0$ of a metric in general relativity? Note that this is not the same questions as the geometric meaning of $R_{\mu\nu}=0$ which has been asked ...
0
votes
0answers
29 views

Uniqueness of the Einstein tensor

This is related with an exercise 17.4-a in MTW Here what i want to show is the Einstein tensor $G_{\alpha\beta} = R_{\alpha\beta} - \frac{1}{2} R g_{\alpha \beta}$ is the only second-rank, symmetric ...
1
vote
2answers
83 views

Derivation of the Riemann tensor confusion

I'm trying to understand the derivation of the Riemann curvature tensor as given in Foster and Nightingale's A Short Course In General Relativity, p. 102. They start by giving the covariant derivative ...
0
votes
1answer
37 views

Deduce the 3 dimensional Anti de Sitter space from the 4 dimensional case [on hold]

From $$AdS^4=\big\{(u,w,x,y,z) \in \mathbb{R} | -u²-w² +x² + y² + z²=-1 \big\}$$ parametrized by $$f(t,\rho, \theta, \phi) = (\sin t \cosh \rho, \cos t \cosh \rho, \sinh \rho \cos \theta, \sinh \rho ...
0
votes
0answers
39 views

Background field expansion in normal coordinates

Background field expansion following form $Y= X+\pi$ where $X$ is my background field and $\pi$ is the fluctuation. From the Normal coordinates we have the expansion of $\pi^{\mu} = ...
2
votes
0answers
30 views

“Projection of metric” vs. “projection of curvatures” [migrated]

Suppose we have a submanifold $M^n$ which is embedded in manifold $M^{n+2}$ and $g_{\mu \nu}$ denotes the metric of $M^{n+2}$. We know that the induced metric on the submanifold is defined by ...
2
votes
1answer
53 views

Is this covariant derivative identity true?

Trying to work through a textbook derivation of the geodesic deviation equation, I've calculated this identity:$$u_{;\beta}^{\alpha}u_{\alpha}=u_{\alpha;\beta}u^{\alpha}.$$ If this is true, I'm ...
1
vote
0answers
41 views

Examples of warped product manifolds?

Bishop and O'Neil defined warped product manifolds. Space-times are good examples of such warped product manifolds. Is there a famous and important example of space-times $I×M$ where $M$ is itself a ...
2
votes
0answers
52 views

Is it possible to build a tensor with the following properties? [closed]

I am searching for a tensor in 4-dimensional space-time with two indices that satisfy: \begin{eqnarray} M_{;\mu }^{\mu \nu } &=&0 \\ M^{\mu \nu } &=&-M^{\mu \nu } \nonumber \\ ...
4
votes
1answer
70 views

Total derivative in action of the field theory

Consider a classical field theory. When applying the least action I see that a term is considered total derivative. We say that $$\int \partial_\mu (\frac {\partial L}{\partial(\partial_\mu ...
1
vote
1answer
90 views

About Christoffel symbols in Riemann normal coordinates

According to the answer to this post, the Christoffel symbols in Riemann normal coordinates are approximated by $$\Gamma^{k}_{ij}(x)~\sim~\frac{1}{2} R^k{}_{ilj}(x_0) \xi^l \tag{5.10}$$ which came ...
0
votes
1answer
72 views

Covariant derivative of stress-energy tensor for a scalar field [closed]

In order to prove that $$\nabla ^\mu T_{\mu\nu} =0$$ I want to find the covariant derivative of $$T_{\mu\nu} = \partial_\mu\phi \partial_\nu \phi -\frac{1}{2}g_{\mu\nu}(g ...
1
vote
0answers
38 views

Conditions for a diagonal induced metric?

Let $M$ be a manifold of dimension $n$ with a (say Lorentzian) metric $g$, that is diagonal in some choice of local coordinates. Let $S$ be manifold of dimension $k<n$ , embedded in $M$ by some ...
3
votes
1answer
141 views

Susy QM and Atiyah-Singer index theorem

Consider maps $t\mapsto x^i(t)$ from circle to some Riemannian (spin) manifold and lagrangian $$ \mathcal L = \frac12 g_{ij}(x) \partial_t x^i \partial_t x^j + \frac12 g_{ij} \psi^j \left(\delta^i_k ...
2
votes
2answers
57 views

Covariant derivative applied to a vector vs. applied to a matrix?

I know there are (say) two different definitions/representations of the covariant derivative: one is the covariant derivative applied to a vector $F$, which reads as $$DF=\partial F+iAF$$ ...
11
votes
4answers
336 views

Is partial derivative a vector or dual vector?

The textbook(Introduction to the Classical Theory of Particles and Fields, by Boris Kosyakov) defines a hypersurface by $$F(x)~=~c,$$ where $F\in C^\infty[\mathbb M_4,\mathbb R]$. Differentiating ...
4
votes
1answer
314 views

Angular momentum in curved spacetime

It is known that the angular momentum components are also a representation of the $SU(2)$ generators. Given a non-trivial spacetime, say a black hole of some kind or AdS space, how can one define the ...
0
votes
1answer
78 views

What is Laplace operator of Schwarzschild-Spherical coordinates? [closed]

This is the Laplace operator of Spherical coordinates: What is the Laplace operator of Schwarzschild-Spherical coordinates? where the Differential displacement of Schwarzschild-Spherical ...
3
votes
1answer
115 views

Correct tetrad index notation

There seems to be some different conventions on the indexes of the tetrad. I am wondering which is the standard, which is correct, and which is an abuse of notation. In Sean Carroll's notes and in ...
4
votes
1answer
196 views

What is discrete phase space?

I've been reading a little about the usual, continuous Wigner functions and phase space quasi-distributions in general, and I believe I understand the idea behind them. The Wigner function arises when ...
1
vote
1answer
98 views

How can we derive the gauge field Lagrangian?

I learned the gauge field Lagrangian is given in this form: $$\mathcal{L} = -\frac{1}{4} \mathrm{Tr}(F_{\mu \nu}F^{\mu \nu}).$$ But how one can derive this equation starting from defining the ...
1
vote
1answer
58 views

One step in deriving the Einstein-Hilbert action

In this amazing first principles derivation of the Einstein-Hilbert action there is one small manipulation needed to show $$c g_{ab,cd}\left(\eta^{ac}\eta^{bd} - \eta^{ab}\eta^{cd}\right)$$ is ...
5
votes
1answer
286 views

Geodesic deviation equation - why does the ordinary second derivative give the correct answer?

I've calculated the correct answer to my problem, but don't understand one of the assumptions I made when doing so. I used the geodesic deviation equation ...
6
votes
0answers
69 views

Is there a null incomplete spacetime which is spacelike and timelike complete?

Geodesic completeness, the fact we can make the domain of the geodesic parametrized with respect an affine parameter the whole real line, is an important concept in GR. Especially, because the lack of ...
7
votes
1answer
116 views

Why is it so coincident that Palatini variation of Einstein-Hilbert action will obtain an equation that connection is Levi-Civita connection?

There are two ways to do the variation of Einstein-Hilbert action. First one is Einstein formalism which takes only metric independent. After variation of action, we get the Einstein field equation. ...
1
vote
2answers
86 views

Good Fiber Bundles reference for Physicists

I'm a student of Physics and I have interest on the theory of Fiber Bundles because of the applications they have in Physics (gauge theory for example). What are good books to learn the theory of ...
0
votes
0answers
62 views

Why doesn't this proof change indices?

In this pdf, in the second line of the proof, $\sigma$ was plugged in where it appears as $$\frac{\partial x^\sigma}{\partial y^{\rho'}}$$ Meanwhile in converting the coordinates of $g^{\mu'\rho'}$, ...
1
vote
1answer
49 views

Christoffell symbols manipulations [closed]

Why is it that $$\Gamma^\lambda_{\lambda\tau}\Gamma^\tau_{\mu\nu} = 0?$$ The same goes for $$\Gamma^\lambda_{\nu\tau}\Gamma^\tau_{\mu\lambda} $$which was set equal to zero by the author..
4
votes
0answers
52 views

Non-trivial scalar quantity

Is there any scalar quantity made of only the Christoffel symbols, determinant of a metric and tensors, not derivatives? In other words, can we construct a scalar quantity which cannot be written in ...
1
vote
2answers
121 views

What is $V^\mu$ if $\nabla_{\mu} V^{\mu}$=scalar?

Suppose there is a quantity written as $\sum\limits_\mu \nabla_\mu V^\mu$ which is invariant under a coordinate transformation, i.e. scalar, where $V^\mu=(V^0,V^1,V^2,V^3)$ and $\nabla_\mu$ is a ...
4
votes
0answers
41 views

What is the status of gauged gravity [duplicate]

The Standard Model of elementary particles is a gauge theory with gauge group $SU(3)\times SU(2)\times U(1)$, which is really a successful theory. We might be able to quantize gravity similarly. ...
4
votes
2answers
134 views

Geometric meaning of parallel transport

The definition of parallel transport of a vector $v^b$ along a curve $C$ with tangent field $\it{t}^a$ is given by Wald's GR as $$t^a \nabla_a v^b = 0$$ Is it correct to think of $\nabla_a v^b$ as ...
1
vote
1answer
32 views

Tensors applied to vector and dual vector fields in GR

This is a specific question about tensor manipulation in Wald's GR. I'm asking for clarification of a trivial step, because I'm working through the text outside the context of a class, without prior ...
3
votes
0answers
46 views

How to derive the cigar soliton solution to the Ricci flow equation? [closed]

I am trying to derive the cigar soliton solution to the Ricci flow equation. Such solution has the form $$ {\frac {{{\it dx}}^{2}+{{\it dy}}^{2}}{{{\rm e}^{4\,t}}+{x}^{2}+{y}^{2 }}} $$ I am ...
4
votes
1answer
78 views

Two expressions for topological instanton number

I have begun to study instantons and I have the following difficulty: $\newcommand{tr}{\operatorname{Tr}}$ I am considering theory with $SU(2)$ gauge group: $S=\frac{1}{2g^{2}}\int \tr ...
0
votes
3answers
120 views

Calculating the Riemann tensor for a 3-Sphere

I have worked out all the connection symbols for the 3-sphere using calculus of variations, cf. this Phys.SE post. So to find the Riemann tensor I am trying to find all the nonzero components of: ...
0
votes
0answers
29 views

Relation between the curvature of a manifold and the number of covariantly constant vector fields that it admits

Suppose that on a four dimensional manifold we are able to explicitly construct four linearly independent covariantly constant vector fields $K^a_{\mu}$: $$D_{\mu}K^a_{\nu}=0,$$ $a=1,2,3,4$ then it ...
1
vote
1answer
61 views

Does the metric define a Riemannian Manifold?

Does a Riemannian Manifold's metric tensor $g$ completely define the manifold, or are more parameters required?
2
votes
3answers
88 views

Generators of the Diffeomorphism Group

So what are the generators of a Diffeomorphism Group? For simplicity, let's consider $ Diff(R^2) $ (diffeomorphisms of the euclidean plane.) Diffeomorphisms are differentiable, invertible ...
6
votes
2answers
293 views

Finding 3-Sphere Christoffel connection coefficients using variational calculus, Sean Carrol problem

I have A 3-Sphere with coordinates $x^{\mu} = (\psi,\theta,\phi)$ and the following metric: \begin{equation} ds^2 = d\psi^2 + \text{sin}^2\psi(d\theta^2 + \text{sin}^2\theta d\phi^2) \end{equation} ...
6
votes
1answer
96 views

Does nature of singularity in black hole depend on material that fell in?

Electromagnetic waves have a tracesless stress energy tensor, and therefore if they are the only fields in a region of spacetime, the Ricci curvature scalar $R=0$ according to GR. However $R^{\mu\nu} ...
3
votes
0answers
52 views

Classical toy models of particles with intrinsic spin

Related to my question here (spacetime torsion, the spin tensor, and intrinsic spin in einstein cartan theory), I'd like to be able to put test particles on a manifold with non-zero torsion and see ...
2
votes
4answers
168 views

Why does a bubble take a spherical shape?

I suspect this has something to do with thermodynamics and the isoperimetric inequality and I'm interested in a mathematical derivation of this result.