4
votes
2answers
90 views

Why is the phase space a symplectic manifold rather than a manifold with a metric?

Why does phase space require a symplectic geometry rather than a metric? Is there some scenario where a metric is unable to describe the notion of length in phase space, specifically in relation to ...
2
votes
1answer
82 views

Can these two terms cancel out?

In trying to prove that $$\Gamma_{\mu\nu\lambda} = \eta_{ab}J^a_bJ^b_{\nu\lambda}.$$ The author canceled out while expanding the first equation $$J^a_{\mu\lambda}J^b_\nu$$ with ...
1
vote
0answers
64 views

Help with parametrization of surface if I'm given the metric [closed]

I've got a homework question. Consider a 2 dimensional space with metric $$ ds^{2} = \frac{dr^{2}}{1 -\frac{2}{r} } + r^{2}d\theta^{2} .$$ I need to show that this is the induced metric ...
0
votes
0answers
77 views

Geodesic deviation on a unit sphere

Very little interest in the original version of this question so I've rejigged it hoping for a more positive response. I'm trying to use the geodesic deviation ...
1
vote
0answers
28 views

Supergravity solution, metric for the total space, and connection

In supergravity solutions, one sometimes encounters the case where the manifold may be a bundle over some base space, and one has to write down the explicit metric regarding such bundle. I would like ...
1
vote
1answer
91 views

Geodesic curvature and Weyl transformations

The geodesic curvature is given by $$k=\pm t^a n_b\nabla_a t^b,$$ where $t^a$ is a unit vector tangent to the boundary of the string worldsheet and $n_a$ is an outward vector orthogonal to $t^a$. I ...
2
votes
0answers
20 views

Using Polyakov-Alvarez Anomaly Formula [closed]

Take $\Sigma=\mathbb{D}$ to be the unit disk with metric $g=\frac{4}{(1+|z|^2)^2}\,|dz|^2$. If $\phi$ is a nice enough function on $\mathbb{D}$, then I want to compute $$\int_{\partial \Sigma} k_g ...
1
vote
0answers
65 views

How to test that a flat metric represents a global three-torus geometry

When introducing Robertson-Walker metrics, Carroll's suggests that we consider our spacetime to be $R \times \Sigma$, where $R$ represents the time direction and $\Sigma$ is a maximally symmetric ...
-2
votes
1answer
62 views

Length in polar coordinates

Say we are in 3 dimensions and use $(-++)$. If we have the metric $$ds^2=-dt^2+dr^2+r^2df^2(t),$$ then what is the third coordinate if the first two were $t$ and $r$? $$X^iX_i=-t^2+r^2+?$$
0
votes
1answer
47 views

The commutator of Killing vectors

I'm going over an assignment for my general relativity course. My solution to the question below strikes me as too short, considering that it appeared in the "longer questions" section of the ...
4
votes
2answers
188 views

Metric tensor in special and general relativity

I'm having trouble understanding the metric tensor in general relativity. What I've understood so far has come from my course lecture notes used in conjunction with "The Road to Reality" by Roger ...
1
vote
0answers
43 views

Laplacian in tensor [closed]

Find $\vec \nabla^2\phi $ when $$ds^{2}=-dt^{2}+a^{2}(t)[dx^{2}+dy^{2}+dz^{2}] $$ or $$g_{ij}=\begin{bmatrix} -1 & 0 &0 &0 \\ 0 &a^{2}(t) &0 &0 \\ 0&0 ...
3
votes
1answer
82 views

Ricci flat compact manifold with $U(1)\times{}SU(2)\times{}SU(3)$ isometry group?

As the title says, is it possible to have a Riemannian Ricci flat compact manifold with $U(1)\times{}SU(2)\times{}SU(3) $ isometry group?
2
votes
1answer
202 views

How to properly construct the electromagnetic tensor in curved space-time?

How do I properly construct the electromagnetic tensor in curved space-time? I have my curved spacetime metric $(+,-,-,-)$ and my magnetic vector potential $A$. I tried two ways but not sure which is ...
2
votes
1answer
102 views

Computing the Christoffel symbols with the geodesic equation

I would like to compute the Christoffel symbols of the second kind using the geodesic equation. To practice, I have tried the Schwarzschild Ansatz $$ g_{00} = \mathrm e^\nu,\quad g_{11} = - \mathrm ...
1
vote
1answer
72 views

Has a metric formulation of electromagnetism ever been attempted? [duplicate]

I understand that electromagnetic fields carry energy, and this energy curves spacetime gravitationally. That's not my question. I'm asking if anyone has tried to formulate electromagnetism in such ...
4
votes
4answers
162 views

What makes a coordinate curved?

Bear with me while I try to explain exactly what the question is. The question Can a curvature in time (and not space) cause acceleration? is imagining a coordinate system in which the curvature is ...
3
votes
0answers
36 views

Sign convention with the $AdS$ metric

One would say that $AdS_n$ satisfies the equations for the scalar curvature (R) and Ricci tensor ($R_{\mu \nu}$), $R = - \frac{n(n-1)}{L^2}$ and $R_{ab} = - \frac{n-1}{L^2}g_{ab}$. But do the signs ...
3
votes
0answers
48 views

Why is the Taub-NUT instanton singular at $\theta=\pi$?

Consider the following metric $$ds^2=V(dx+4m(1-\cos\theta)d\phi)^2+\frac{1}{V}(dr+r^2d\theta^2+r^2\sin^2\theta{}d\phi^2),$$ where $$V=1+\frac{4m}{r}.$$ That is the Taub-NUT instanton. I have been ...
1
vote
3answers
125 views

Why do we need a metric to define gradient?

For me, the gradient of a scalar field (say, in three dimensions) is simply (formally) $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\frac{\partial f}{\partial z} ...
2
votes
2answers
67 views

Orthogonality in curved space/spacetime

When are two vectors orthogonal in curved spacetime? From wikipedia: "In 2-D or higher-dimensional Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they ...
10
votes
4answers
147 views

Difference between matrix representations of tensors and $\delta^{i}_{j}$ and $\delta_{ij}$?

My question basically is, is Kronecker delta $\delta_{ij}$ or $\delta^{i}_{j}$. Many tensor calculus books (including the one which I use) state it to be the latter, whereas I have also read many ...
2
votes
0answers
40 views

Laplacian in 4 spatial dimensions; 4th dimension warped

How can I prove the form of the Laplacian in four spatial dimensions, using the identification $y = y + 2\pi R$ for the fourth dimension and assuming the others as the usual Cartesian ones? I want to ...
11
votes
4answers
1k views

Why is the space-time interval squared?

The space-time interval equation is this: $$\Delta s^2=\Delta x^2+\Delta y^2+\Delta z^2-(c\Delta t)^2$$ Where, $\Delta x, \Delta y, \Delta z$ and $\Delta t$ represent the distances along various ...
2
votes
4answers
183 views

GR matter-free equations and Schwarzschild geometry

I am reading some lecture notes on General relativity (undergraduate level) and I do not understand a sequence of statements about the topics in the title. After stating that the for matter-free ...
2
votes
2answers
54 views

Show that two families of curves are orthogonal (without using orthogonal trajectories)

I'm reading through Hartle's General Relativity and came across this question: Consider the following coordinate transformation from rectangular coordinates $(x,y)$, labeling points in the plane ...
2
votes
1answer
97 views

Calculate divergence of vector in curvilinear coordinates using the metric

In a curved $(3+1)$ dimensional spacetime with metric components $g_{\mu \nu}$, the covariant derivative of a $4$ vector $\mathbf V = (V^0, \vec V)$ is given by $$\nabla_\mu~ V^\mu = ...
14
votes
4answers
411 views

The Lagrangian as a metric

My question is, can the (classical) Lagrangian be thought of as a metric? That is, is there a meaningful sense in which we can think of the least-action path from the initial to the final ...
0
votes
0answers
61 views

Why such hypersurface orthogonal vector leading to $g_{0i}=0$ for $i=1,2,3$?

Suppose that the hypersurface orthogonal co-vector $W$ us perpendicular to the family of hypersurface defined by a function $\varphi$ with $\varphi=constant$. If we choose a coordinate in which ...
1
vote
0answers
103 views

Covariant Derivative with a Torsion Free Metric

Where $\triangledown$ is the covariant derivative and we are to assume that the connection is torsion free (that is, we can exchange the lower indices of the connection coefficients), how can I prove ...
4
votes
1answer
164 views

Computing Curvature via Cartan Formalism

Given a metric $g_{\mu \nu}$, one can select an orthonormal basis $\omega^{\hat{a}}$ such that, $$ds^2= \omega^{\hat{t}}\otimes\omega^{\hat{t}} - \omega^{\hat{x}} \otimes \omega^{\hat{x}} - ...$$ By ...
2
votes
3answers
431 views

D'Alembertian for a scalar field

I have read that the D'Alembertian for a scalar field is $$ \Box = g^{\nu\mu}\nabla_\nu\nabla_\mu = \frac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}\partial^\mu). $$ Exactly when is this correct? Only for ...
0
votes
4answers
151 views

How to determine “timelike”-ness without using a coordinate system?

It has been stated here that: we can say, without introducing a coordinate system, that the interval associated with two events is timelike, lightlike, or spacelike. This assertion appears at ...
0
votes
0answers
92 views

Curvature based derivation of Schwarzchild Metric

I'm a third year maths undergrad and I'm trying to find (and follow) a curvature based derivation of the Schwarzchild metric, if there exists such a proof?
1
vote
0answers
55 views

Contracting Indices in General relativity [duplicate]

I was reading a book about general relativity and I came across these two equations $$ \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{\Gamma}^{\mu}_{\sigma\rho}+ ...
9
votes
1answer
311 views

Contracting Indices

Does anyone know how to get from (1) to (2) in the system $$ \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{{\Gamma}}^{\mu}_{\sigma\rho}+ ...
2
votes
1answer
71 views

Finding the components of the Riemannian tensor given the components of a metric

I am looking at a manifold of dimension $n$ (And I am considering a local co-ordinates system $x_1,x_2,\ldots x_n$) and the metric defined by the components $g_{ij} = \frac{\delta_{ij}}{x_1^2}$. I'm ...
3
votes
1answer
132 views

Schwarzschild geodesics

I've found on Wikipedia that energy $E$ and angular momentum $L$ of a particle are conserved quantities in Schwarzschild metric. It's written: $$L=mr^2 \frac {d\phi} {d\tau},$$ ...
0
votes
0answers
73 views

What is the physical meaning of the Eddington - Finkelstein metric?

I want to see a some physical process (experimental) that could explain the many transformations of coordinates into this mathematical procedure. (really two transformations, but i think that is a ...
4
votes
1answer
110 views

Extent of coordinate freedom to set metric components along a spacetime path

If we describe spacetime with a Lorentzian manifold, it is always possible to choose a coordinate system such that at any particular point $x^\alpha$, the components of the metric are: $$ ...
2
votes
1answer
125 views

How to prove that zero Weyl tensor predicts no deflection of light?

There is Nordstrom theory, which can be given as $$ C_{\mu \nu \alpha \beta} = 0. $$ The solution of Einstein equations for this case is conformally flat metric: $$ g^{\mu \nu} = e^{\epsilon \varphi ...
2
votes
2answers
174 views

How does the Einstein Equivalence Principle imply a spacetime with a metric (and a connection)?

I have at hand the book by Clifford Will, "Theory and Experiments in Gravitational Physics", and the following Living Reviews in Relativity article. He quotes the Einstein Equivalence Principle (EEP) ...
2
votes
2answers
164 views

Why do we must know the Weyl tensor for 4-dimensional space-time?

I heard that we must know the Weyl tensor for fully describing the curvature of the 4-dimensional space-time (in space-time with less dimensions it vanishes, so I don't interesting in cases of less ...
4
votes
0answers
352 views

How to prove that Weyl tensor is invariant under conformal transformations?

I need to verify that the solution for vanishing Weyl tensor is conformally flat metric $g_{\mu\nu} = e^{2\varphi}\eta_{\mu\nu}$. The most convenient way to show this is to prove that Weyl tensor is ...
0
votes
1answer
592 views

General Relativity: Christoffel symbol identity

I want to show that $$\Gamma ^{\mu}_{\mu \nu}=\partial _\nu (\ln \sqrt{|g|}) .$$ (Here $|g|$ denotes the determinant of the metric.) Working out the left hand side:\begin{align} \Gamma ^{\mu}_{\mu ...
0
votes
2answers
461 views

General relativity: Induced metric and Killing vector fields

Assume that in spacetime ($M,g_{ab}$) there is a hypersurface generated by a set of independent one-parameter transformations acting on one single point, the generators of these transformations being ...
2
votes
2answers
317 views

Weight of a tensor density

Is there any freedom in choosing the weight of a tensor density? I have seen in some papers that they introduce a tensor density made from metric with a special weight. There is a tensor density with ...
3
votes
1answer
140 views

Index raising and lowering - how does it work?

In the context of four-dimensional spacetime, how does the metric turn a tangent vector into a gradient, and vice versa? By this I mean that I know the metric can be used to raise and lower indices: ...
2
votes
1answer
507 views

How to derive the metric for a 2-sphere

I have a question in Polchinski's string theory vol I p 167. It is said For example, $$ds^2= \frac{ 4 r^2 dz d \bar{z} }{(1+ z \bar{z})^2} = \frac{ 4 r^2 du d\bar{u}}{ (1+ z \bar{z})^2} ...
-1
votes
1answer
220 views

Metric tensor in General Relativity or otherwise [closed]

What is the metric tensor? How can this be a covariant and contravariant tensor, or a mixed tensor, by raising and lowering indices? How it relates to distance function (metric) and angles? How ...