The variables used in general relativity to describe the shape of spacetime. If your question is about metric units, use the tag "units", and/or "si-units" if it is about the SI system specifically.

learn more… | top users | synonyms (2)

4
votes
2answers
146 views

Getting the Lagrangian from the action in curved spacetime

Suppose I have this action: $$ S = \int \mathrm d^4 x\sqrt{-g}\times \text{something}$$ where $g$ is the determinant of the metric. Should I take the Lagrangian to be: $$ \mathcal L = \sqrt{-g} ...
0
votes
0answers
103 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
2answers
131 views

Inner products in relativity

In physics, the definition of a dot (inner) product is often between a vector (“contravariant vector”) and a covector (“covariant vector”). However, in mathematics, a dot product is always defined ...
2
votes
2answers
135 views

Affine connection notation

Can ${g}^{\mu\sigma}{\Gamma}^{\rho}_{\sigma\nu}$ be written as ${\Gamma}^{\mu\rho}_{\nu}$? If so how come this symbol never appears in any GR book?
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
344 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}+ ...
1
vote
0answers
180 views

Understanding spherically symmetric metric

In these lecture notes the static isotropic metric is treated as follows (p. 71): Take a spherically symmetric, bounded, static distribution of matter, then we will have a spherically symmetric ...
1
vote
1answer
86 views

What do they mean with: photon scattering with $q^2=-Q^2\leq 0$

In a scattering problem, let q denote the four-momentum of the photon. Is $q^2=-Q^2\leq 0$ simply a statement of what metric one uses and simultaneously a definition of $Q^2$?
2
votes
1answer
106 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 ...
1
vote
0answers
100 views

Ricci scalar higher dimensions

I was wondering if there is a straightforward way to compute the Ricci curvature of a metric that has the form (à la Kaluza-Klein): $g_{MM}\equiv\begin{pmatrix}g_{\mu\nu}&g_{\mu ...
3
votes
0answers
230 views

Squashed 3-sphere?

What is a squashed 3-sphere? In context of quantum gravity. I stumbled upon a term 'squashed 7 sphere' but that's concerning supersymmetry. Is it just normal 3-sphere metric, that is just 'squashed' ...
2
votes
1answer
162 views

Why are there two metric signature conventions?

I understand that it is more common in GR for the metric to be given a $(-,+,+,+)$ signature and more common in particle physics (or field theory, as Peskin & Schroeder tells me) to use the ...
0
votes
1answer
77 views

One particle near two Schwarzschild black holes

I have a particle near two Schwarzschild black holes. Let the black holes remain at rest so that only the particle is moving for the observer. We are in a plane. I calculate the distance travelled by ...
2
votes
1answer
536 views

Variation of modified Einstein Hilbert Action

In general relativity one can derive the Einstein Field Equations by the principle of least action through variations with respect to the inverse of the metric tensor. In some modified theories of ...
1
vote
0answers
86 views

The time dilation in an oscillating elevator

Suppose you are in an elevator which oscillates vertically with a frequency $\nu$. How will we find the time dilation in this oscillating reference frame ? If the lift is accelerating upward or ...
3
votes
1answer
158 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: $$L=mr^2 \frac {d\phi} {d\tau},$$ ...
1
vote
4answers
347 views

Space-like and time-like: where do the names come from?

Space-like separated events are events that, in a well-chosen reference frame, can take place at the same time but never happen at the same location. On the other hand for time-like events, one can ...
5
votes
1answer
503 views

Step by step algorithm to solve Einstein's equations

I cannot completely understand what is a regular method to solve Einstein's equations in GR when there are no handy hints like spherical symmetry or time-independence. E.g. how can one derive ...
1
vote
2answers
354 views

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

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 ...
5
votes
1answer
144 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: $$ ...
1
vote
3answers
174 views

Can General Relativity Metric Tensor be independent of a particular co-ordinate index in a local area?

For example in a particular local area, can the metric tensor be totally independent of $z$ co-ordinate in $(t,x,y,z)$ co-ordinate system? This way the distance function will not contain $z$ ...
2
votes
1answer
149 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 ...
1
vote
2answers
298 views

Metric Expansion Of Space

I just do not understand this concept of metric expansion of space. Shouldn't the galaxies move away from each other. How can the space between them expand if the galaxies are not moving away from ...
1
vote
0answers
44 views

Allowed transformations in General Relativity [duplicate]

So in Special Relativity we have: $$ \Lambda \eta \Lambda^T=\eta $$ Is there an analagous formula for the metric in General Relativity?
1
vote
1answer
114 views

Best way to check for anisotropy given a metric tensor

Carroll gives the definition of isotropy at a point as given vector $V$ and $W$ in $T_{p}M$, there is some isometry that can push $V$ forward such that it ends up parallel to $W$. I understand what ...
1
vote
1answer
150 views

Are there any good references on the “gravitational” curvature of spacetime of a moving mass being distorted due to special relativity?

In this Wikipedia paragraph suggesting an explanation for the phenomenon of inertia, it claims: Another physicist, Vern Smalley, has derived the Lorentz transformation for mass by assuming that ...
1
vote
1answer
214 views

Relation between symmetries and Killing vectors by Weinberg

In his book, "Gravity and Cosmology", Weinberg talks about relations between homogeneous metric spaces and Klling vectors. First he says about infinitesimal isometrics $$ x^{\alpha}{'} = x^{\alpha} + ...
2
votes
1answer
91 views

What does it mean for a metric to be regular?

A problem in Carroll (a general relativity textbook) asks if a certain metric is regular. What does it mean for a metric to be regular?
2
votes
2answers
237 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) ...
3
votes
1answer
388 views

What's the importance of conformal transformations in general relativity?

I tried to understand the importance of conformal transformations in general relativity, but I failed. I didn't see that conformal transformations help to simplify the metrics, and also I didn't see ...
1
vote
2answers
335 views

How to prove the raising/lowering indices operation?

I've read this related question, though it didn't satisfy me; I hope this complements it. I know that if I contract a covariant tensor ${A_{\alpha\beta}}$ with a vector ${B^\beta}$, I get some other ...
4
votes
1answer
146 views

Some hints for special case of metric tensor in GR

Let's have metric $$ ds^2 = dt^2 - dx^2 - dy^2 - dz^2 - 2f(t - z, x, y)(dt - dz)^2. $$ I need to prove that it is an exact solution for Einstein equations in vacuum for $\partial_{x}^{2}f + ...
2
votes
2answers
228 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
501 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 ...
6
votes
2answers
1k views

Infinitesimal Lorentz transformation is antisymmetric

The Minkowski metric transforms under Lorentz transformations as \begin{align*}\eta_{\rho\sigma} = \eta_{\mu\nu}\Lambda^\mu_{\ \ \ \rho} \Lambda^\nu_{\ \ \ \sigma} \end{align*} I want to show that ...
3
votes
2answers
361 views

The signature of the metric and the definition of the electromagnetic tensor

I've read the definition of the electromagnetic field tensor to be ...
1
vote
0answers
134 views

Linearized gravity and symmetries

I have naive question. When we analyzing weak gravity field we introduce expression for metric tensor as $$ g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}, \quad \eta_{\mu \nu} = diag(1, -1, -1, -1), ...
2
votes
1answer
329 views

Ricci scalar in Scalar Field in Curved Space-time

I was recently looking at a Lagrangian of a scalar field in curved space-time at http://www.unc.edu/~mgood/research/Carroll_QFT_CS.pdf on page 8. I am not a physicist, and I am currently studying ...
2
votes
1answer
259 views

Cosmological metric with off-diagonal terms?

In the context of Cosmology models, What are examples of metrics with off-diagonal terms?
0
votes
0answers
73 views

Angular Momentum with Upper Index

I am asked to show $[L^2,L_i] = 0 $, but with the definition : $L^2 \equiv L_i L^i$ I tried this: $[L_i L^i,L_i] = L_i [L^i,L_i] + [L_i,L_i]L^i$ We know that : $[L_i,L_i]$ = 0 , so we have, $[L_i ...
1
vote
1answer
1k 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
542 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 ...
1
vote
1answer
33 views

Flat space current conservation sign confusion

It is said that in Minkowski spacetime, the current conservation law for the number current $N^\mu$ where $N^0$ is the number density and $N^i, i=1,2,3$ is the particle flux in the $x^i $ direction, ...
3
votes
0answers
139 views

About Dirac equation in curved spacetime (spherical)

I would like to ask you about the separation of variables of the Dirac equation in curved space-time. The metric is given by $$ds^{2}=-dt^{2}+dr^{2}+r^{2}d\theta^{2}+\alpha^{2}r^{2}\sin^{2}\theta ...
1
vote
3answers
235 views

Coordinate transformations of the metric tensor

Let's have metric (it describes the space-time of uniformly accelerating observer in Minkowski space-time) $$ ds^2 = v^2du^2 - dv^2. \qquad (.0) $$ I need to find expressions for $u = f(x, t), v = ...
2
votes
2answers
373 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 ...
2
votes
2answers
137 views

Distinguish between Past and Future

When writing the metric in Minkowski space, how can we distinguish between the past and the future? I understand the answer after drawing the light cone but I want to know how we get that by just ...
2
votes
1answer
437 views

Lightcone coordinates

The Light cone coordinates are defined as $$x^+ = x^0 + x^3$$ $$x^- = x^0 - x^3$$ Then in the light cone coordinates the position 4-vector becomes: $(x^+, x^-, x^1, x^2)$ . Zwiebach in his A First ...
3
votes
3answers
1k views

How the Lorentz transformation affects the metric tensor?

After performing a Lorentz transformation, the orthogonal coordinates will become askew, as in the following figure: and in such coordinate system, according to this Wikipedia article, the metric ...
3
votes
1answer
161 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: ...