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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Why do the Einstein field equations (EFE) involve the Ricci curvature tensor instead of Riemann curvature tensor?

I am just starting to learn general relativity. I don't understand why we use the Ricci curvature tensor. I thought the Riemann curvature tensor contains "more information" about the curvature. Why is ...
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Particle motion characteristic

I'm making a particle motion raffling normal numbers. The normal random numbers raffled are the angles of the directions that the particle is going. The particle speed is constant. Look how this is ...
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Gravitational time dilation in changing curved space time

Imagine a portion of spacetime which is changing its spacetime curvature because of an object with great mass travelling nearby. For instance, before it was flatter, and after the object passes it ...
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Curvature gravity and a falling apple? [duplicate]

I know very little of physics after Einstein. I am aware of that Einstein's gravity theory says that the existence of matters creates curvature of a space-time, so that our Earth orbits our Sun. I ...
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44 views

Are Laplace Operator and mean curvature exactly the same thing for 2D function?

Let's assume we study 2D function/surface f(x,y). Then Laplace Operator is defined as: $$\nabla^2 f=\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}$$ And the mean curvature: let ...
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General relativity: is curvature of spacetime really required or just a convenient representation?

I'm not really far into the general theory of relativity but already have an important question: are there formulations that can do without spacetime curvature and describe the general theory of ...
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Curvature of spacetime as a real thing?

I get the curvature tensor in General Relativity, it is “just” math. Does space-time REALLY curves as a tangible thing, or is Einstein proposing a mathematical abstraction? More naively, please ...
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30 views

Flatness and Kinetic Energy

Why the curvature parameter can be interpreted as the difference between the average potential energy and the average kinetic energy of a region of space? Curvature ...
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Is the surface of a heavy sphere bigger than $4 \pi r^2$ due to general relativity?

I am unfortunately not familiar with the mathematics behind general relativity. However, on a heavy planet (say a sphere) gravity will bend space-time in a way that an object initially in rest, will ...
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How is $\Omega_0 = 1$ when the characteristic “teardrop” past light cone seems to admit curvature?

Introduction: The top graphic is just one I pulled from a page describing the process of detecting cosmic curvature. The second graphic is one I drew up to illustrate my misunderstanding. My ...
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Curved space-time VS change of coordinates in Minkowski space

I'm looking for a rather intuitive explanation (or some references) of the difference between the metric of a curved space-time and the metric of non-inertial frames. Consider an inertial reference ...
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Relationship between mass and the radius of curvature of space and time

What is the relationship between mass and the radius of curvature of space and time created due to the presence of the mass? please give the mathematical relation if there is any?
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If space warps distort moving objects' trajectories, does it mean that static objects are immune to gravity? [closed]

If gravity is just space distortion, which affects trajectories of moving objects, then a static object (not moving, thus no trajectory) will not suffer any type of accelerating force from gravity? ...
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Is there any relationship between the $E=mc^2$ equation and the $a_n=\kappa v^2$ formula for the normal component to acceleration?

To clarify, I know very little about physics and don't pretend to have any insight whatsoever into relativity beyond what has entered the popular imagination; my knowledge is more or less at the level ...
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Physical visualisation of curvature

I was wondering-how do you visualise curvature in the context of general relativity. The gravity well and trampoline analogies are quite wrong, so I want a more realistic approach to it (say, the way ...
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How can we see that the Riemann curvature tensor is covariant?

The Riemann curvature tensor, using the conventions of wikipedia, is written in terms of Christoffel symbols as: $$ \tag{1} R^\lambda_{\,\,\mu \nu \rho} = \partial_\nu \Gamma^\lambda_{\,\,\rho \mu} - ...
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Visualizing gravity in 3D

We've all seen the depiction of gravity bending space downwards, and so attracting objects into the dent it creates, cf. e.g. this and this Phys.SE posts. That's intuitive and makes a lot of sense, ...
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If gravitation is due to space-time curvature, how can a body free-fall in a straight line?

According to general relativity, Gravity is due to space-time curvature. Then all paths must be curved. If so, how can there be any straight line motion? The body must follow a curved path. So, there ...
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55 views

Space time curvature bends back

If our perception of space-time curvature is gravitation and Reduced Gravity Plane can reach weightlessness on some point of its trajectory, doesn't that mean that when Reduced Gravity plane reaches ...
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Curvature of a particle move

I'm simulating a particle movement following a normal distribution. How this is done: My particle has a constant speed v and every step the particle move, I ...
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Do the concepts of intrinsic and extrinsic curvatures imply that all spaces are embedded in a higher dimensional space?

The concepts of intrinsic and extrinsic curvature seem to imply that all spaces must be embedded in a higher dimensional space? What does this imply for physical reality?
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How can the universe be flat and have no center if universal mass-energy content is finite?

WMAP measurements confirm that the universe is flat within a 0.4% margin of error. If we assume the universe is flat and there is no 'center' then how could the mass energy content be finite since ...
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Coordinate Symbol confusion in general relativity

In a previous post (Finding the metric tensor from the Einstein field equation?), the equation used lambda, rho mu and nu (not sure of the names of the letters!) for the Ricci tensor and swapped to a, ...
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Gravity is curved geometry: A fact of nature or model-dependent interpretation?

We are regularly taught in high-schools and universities that, according to General Relativity (GR), gravity is nothing but a manifestation of space-time curvature (which, in its turn, is caused by ...
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Interpreting the Kretschmann scalar

How do you interpret the Kretschmann scalar (in general relatvity)? What can you tell from it? The Kretschmann scalar is defined as $$K = R_{abcd} R^{abcd} $$ where $R_{abcd}$ is the Riemann ...
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(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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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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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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31 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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74 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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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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104 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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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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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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131 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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127 views

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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218 views

What made Einstein think that gravity was caused by the curvature of spacetime?

What observation/thought experiment led him to think this?
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0answers
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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
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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
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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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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
45 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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68 views

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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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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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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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 ...