Use this for questions pertaining to curvature of manifolds. Does not need to be specific to general relativity, but also for curvature of e.g. a [tag:calabi-yau] manifold.

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Determination of Ricci tensor and Scalar curvature from vielbeins [migrated]

Consider the following metric: $ds^2=h(r)\bigg(dr^2+r^2\big(d\theta^2+\sin^2\theta ~d\phi^2+(d\psi+\cos\theta ~d\phi)^2\big)\bigg)$ We can try to compute the Ricci scalar of this metric by using ...
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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 ...
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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 ...
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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} = ...
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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 ...
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Most general second-rank symmetric tensor in Einstein theory

I am reading MTW page 407, Exercise 17.1. (a) Show that the most general second-rank, symmetric tensor constructable from Riemann and $g$, and linear in Riemann, is $$a R_{\alpha\beta} + b R ...
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1answer
28 views

Direction of formation of Black Hole

When Black holes are getting formed, in which direction in space they form? For example, I have read that formation of Black Holes is same as forming a hole on a rubber sheet by a spherical ball, so ...
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2answers
55 views

How does a roller coaster car stay on the track on a curve [closed]

Say a roller coaster car is going up a ramp to a drop. At some point it needs to traverse a curve to get to the drop. In general, since the car is constrained to the rail, how is it able to move ...
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2answers
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How is the Ricci scalar $R=0$ here?

Given the metric in the form: $$ds^2 =-A(r)dt^2 +B(r) dr^2 dr^2 +r^2(d\theta ^2 +\sin^2\theta d\phi^2)$$ Papapetrou in his book said that $R=0$ But when I performed it I didn't get zero. For ...
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91 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 ...
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Can hyperbolic space be bounded?

There are many visualisations of hyperbolic geometry using Poincaré disks. What are their purpose? Can hyperbolic space be bounded? Can we endow the disk with the structure described by the FLRW ...
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de Sitter–Schwarzschild metric in the Kretschmann Gravity?

The Kretschmann Gravity (Gauss-Bonnet Gravity,Lovelock Gravity) results when the Ricci scalar is replaced by the Kretschmann invariant in the Lagrangian of the General Relativity. We consider here a ...
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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 ...
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1answer
45 views

Newtonian tidal forces and curvature

Today in my physics class, my lecturer said something which confused me. He said: "Newtonian tidal forces are reinterpreted as a manifestation of curvature in General Relativity". Now I know what ...
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2answers
66 views

Physical meaning of harmonic function?

In complex numbers, we define a harmonic function as a twice continuously differentiable function such that the Laplace operator acting on it gives zero. Can anybody explain me the physical ...
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1answer
98 views

Verifying a solution to Einstein's vacuum field equations

I need to verify a solution to Einsteins vacuum field equations. I have the solution as follows $$ds^2=a\,dt^2+b\,dr^2+\cdots$$ Is the following the right approach? Einsteins equation reduces to ...
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1answer
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Is the Universe still believed to be flat?

I have read a handful of old articles from mid 2013 expressing that the Universe may, in fact, be curved. http://www.nature.com/news/universe-may-be-curved-not-flat-1.13776 ...
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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 ...
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1answer
63 views

Riemannian curvature tensor [closed]

In Einstein's field equations, it includes only energy momentum tensor of the matter alone. However, it doesn't include the energy momentum tensor of the field. In Professor Hamber lectures on General ...
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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: ...
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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 ...
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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?
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1answer
36 views

Error in measuring distance ignoring curvature of Earth [closed]

Suppose you model distance as a flat 2d plane rather than a curved surface. Given that the radius of the Earth is about 6400 km, approximately how far must you travel before the relative error ...
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2answers
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Curvature of electrostatic potential is zero

Could you please expound upon this claim? I found such claim on Zangwill's Classical Electrodynamics, which states that constraint coming from Laplacian equation implies electrostatic potential has ...
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1answer
67 views

Can a lone black hole in a closed Universe evaporate?

If there is a closed Universe which only has a black hole in it, can that black hole evaporate? As the black hole evaporates, it gives off energy, which will eventually come back and be re-absorbed ...
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67 views

Riemann Curvature Tensor Symmetries Proof

I am trying to expand $$\varepsilon^{{abcd}} R_{{abcd}}$$ by using four identities of the Riemann curvature tensor: Symmetry $$R_{{abcd}} = R_{{cdab}}$$ Antisymmetry first pair of indicies ...
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57 views

Equation regarding the Riemann tensor in the Cartan formalism [closed]

I have a problem verifying the following equation (in three dimensions) $$\epsilon_{abc} e^a\wedge R^{bc}=\sqrt{|g|}Rd^3 x$$ where $R$ is the Ricci scalar and $R^{bc}$ is the Ricci curvature ...
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1answer
80 views

Riemann curvature tensor notation in Wald

This question is entirely on tensorial notation in Wald's General Relativity. When specifying the properties of the Riemann tensor on pg39, he states: $R_{[abc]}^{\quad \ \ \ d} = 0$ and For the ...
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1answer
72 views

Space curvature based on net energy = 0

In Neil DeGrass Tyson's epic video, at 2:26:50 ...
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1answer
102 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 ...
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142 views

Tricks for Computing Riemann Curvature Tensor with Levi-Civita connection

I am new to differential geometry, so far it seems to me that computing the Riemann tensor tends to be a rather tedious task, I wanted to know whether there are some tricks that I am missing. In ...
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Spacial curvature and expanding space

If we take the analogy that in an empty space the space is just a flat sheet then if there is a single planet or a star then the flat sheet will curve below the planet leaving a curvature shaped like ...
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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 ...
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Riemann curvature of a unit sphere

The Riemann curvature of a unit sphere is shown in many textbooks to be sine-squared theta where theta is the azimuthal angle of spherical co-ordinates. But what is the significance of the angle and ...
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1answer
64 views

Conservation in space-time curvature

Pardon this possibly naive question. I'm starting to poke around in the topic of General Relativity (as soon as I can pull myself back up out of the vortex of underlying mathematics that I've gotten ...
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Examples of manifolds (not) being: flat, homogeneous and isotropic

I am looking for (at least) one example of the following manifolds: Flat, homogeneous and isotropic Curved, homogeneous and isotropic Flat, non-homogeneous and isotropic Flat, homogeneous and ...
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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 ...
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2answers
118 views

How can space be euclidean when light bends?

I have read people arguing that tridimensional space sections of space time continuum (whatever its number of dimensions) appears to be euclidean from empirical evidence. I cannot reconcile it with my ...
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1answer
61 views

Maximum curvature in a black hole

If the curvature inside the horizon of a black-hole is not infinite in some quantum gravity theories (as in Loop quantum gravity), then what is the expression of the maximum value of the curvature ...
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1answer
52 views

Number of Stars vs Value of Omega (Crtitical Density of the Universe)

I may be badly mixing things up here. If I am, please kindly correct me. As I understand it, if the universe was too dense at the start of the big bang, it would have collapsed back in on itself. Too ...
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3answers
208 views

Space-time curvature creates gravity or is it (could it be) vice-versa too?

Mass (Energy) creates space-time curvature and thus it forms the reason for gravity. Can it be vice-versa too? Like, mass created gravitational field, gravitational field created space-time curvature? ...
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1answer
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Is the apparent lack of (Ricci) curvature in the Schwarzschild metric due to a choice of coordinates?

I've been lightly studying GR lately. Something that has been bothering me has been the lack of (Ricci) curvature produced from the Schwarzschild metric in the few lectures I've watched, as well as ...
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1answer
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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 ...
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4answers
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Why does a flat universe imply an infinite universe?

This article claims that because the universe appears to be flat, it must be infinite. I've heard this idea mentioned in a few other places, but they never explain the reasoning at all.
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4answers
188 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 ...
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2answers
169 views

Can a curvature in time (and not space) cause acceleration?

I realize that the curvature of space-time causes acceleration (gravity). Is it possible to have a curvature only of space, or a curvature only of time? If so, would a curvature only of space, or a ...
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1answer
110 views

The relationship between Lorentz Lie algebra and curvature

Here I transfered the question from the comment The relationship between spin and spinor curvature How $\mathcal{R}_{ab} = \frac{1}{4}R_{abst}\gamma^s \gamma^t$ is from $\Psi \mapsto \Psi + ...
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1answer
169 views

Two definitions of Riemann curvature tensor

I am relatively used to the coordinate free expression of the Riemann tensor: $$ R(X, Y)Z=\nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z - \nabla_{[X, Y]} Z, $$ where $\nabla$ is the Levi-Civita connection ...
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1answer
117 views

How to understand the curvature of this metric?

Suppose we have the metric $ ds^2 = dr^2 + \alpha^2 d\phi^2$, where $\alpha$ is a constant, $0 \leq r \leq \infty$, $ 0 \leq \phi \leq 2 \pi$ and we identify points $\phi = 0$ with points $\phi = ...