3
votes
0answers
39 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 ...
0
votes
3answers
81 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: ...
1
vote
1answer
62 views

Ricci Scalar of the five-dimensiional Reissner–Nordström metric is different to zero?

The Ricci scalar of the four-dimensional Reissner–Nordström metric is equal to zero. In the case of the five-dimensional Reissner–Nordström metric, the Ricci scalar is different to zero?
1
vote
1answer
97 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
25 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
69 views

Higher-Dimensional Metrics in (Hyper)-Spherical Coordinates

I want to compute the components of the Riemann curvature tensor (for a case similar to the Schwarzschild solution) in 4 + 1 dimensions, but I want to use a higher-dimensional analogue of spherical ...
1
vote
1answer
74 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
182 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 ...
4
votes
1answer
173 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 ...
0
votes
0answers
94 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?
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 ...
1
vote
0answers
79 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 ...
2
votes
1answer
127 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
1answer
146 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 ...
4
votes
0answers
362 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 ...
2
votes
1answer
254 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 ...
-1
votes
1answer
224 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 ...
13
votes
1answer
450 views

What are the local covariant tensors one can form from the metric?

Normally in differential geometry, we assume that the only way to produce a tensorial quantity by differentiation is to (1) start with a tensor, and then (2) apply a covariant derivative (not a plain ...
2
votes
1answer
218 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} ...
2
votes
2answers
405 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 ...
1
vote
0answers
523 views

How to calculate Riemann and Ricci tensors for a sphere? [closed]

Let's have the metric for a 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 ...
1
vote
0answers
50 views

How to prove the derive the expression for space part of Riemann tensor for homogeneous and isotropic space-time?

It's not a homework!! For spheric, hyperbolic and flat case $$ dl^{2} = R^{2}\left(d \psi^{2} + sin^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})\right), $$ $$ dl^{2} = R^{2}\left(d ...
3
votes
3answers
484 views

How scalar curvature of following spacetime can be equal to zero?

For an interval of this spacetime, $$ ds^{2} = c^{2}dt^{2} - c^{2}t^{2}(d \psi^{2} + sh^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})), $$ scalar curvature is equal to zero. Also, Ricci ...
0
votes
2answers
85 views

Changing the scalar curvature (k = 0,+1,-1) with coordinate transformations?

I would like to prove that I can (or can't) change curvature of space, k = 0,+1,-1, via general coordinate transformations, which in principle can mix space and time coordinates together.
7
votes
4answers
348 views

How do you tell if a metric is curved?

I was reading up on the Kerr metric (from Sean Carroll's book) and something that he said confused me. To start with, the Kerr metric is pretty messy, but importantly, it contains two constants - ...
3
votes
1answer
254 views

Material strain from spacetime curvature

Let's say that you moved an object made of rigid materials into a place with extreme tidal forces. Materials have a modulus of elasticity and a yield strength. Does the corresponding 3D geometric ...