Questions tagged [curvature]

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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140 views

Why is the right hand side of the Equation of Geodesic Deviation 0?

I'm reading through several primers on GR and I keep seeing the same thing over and over again with no explanation: why is the right hand side of this equation zero: $$\frac {D^2\xi^a}{Du^2}+R^{\alpha}...
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2answers
63 views

Perception of curvature of human eye [closed]

How are human eyes able to detect the different curvatures of surface ? Basically how are the human eyes able to differentiate between a plane surface and a convex or a concave surface?
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1answer
86 views

Are the actual singularities with geodesic incompleteness the same as curvature singularities?

Types of singularities include curvature singularities and conical singularities. So, for a curvature singularity(black hole) with geodesic incompleteness, is it the same as a physical singularity? If ...
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1answer
84 views

General Theory of Relativity [duplicate]

Why is General Theory of Relativity not applicable in the singularity of Black Hole?
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2answers
154 views

Distortion of space and time

What will happen to the distorted space and time around a mass when it is converted into energy? Will it back to its original configuration (0 gravity)? Or space time oscillates? Or anything else
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3answers
171 views

Is gravity a force given that it derives from curved massless space-time?

One of the answers to a similar question regarding gravity concluded that gravity is an "observed effect" of the curvature of space-time. I read this (and other answers) to imply that gravity results ...
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2answers
107 views

Unable to Understand a Step in Wald's General Relativity book

Refer to the following picture: How can one jump from ($3.2.30$) to $(3.2.31)?$
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2answers
245 views

Why does the lowered Riemann tensor only have 20 independent components for the Schwarzschild metric?

I have seen quite a bit online about how there are only 20 independent components for the (lowered) Riemann tensor $R_{abcd}$ for the Schwarzschild metric. I've been told this follows from the ...
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3answers
74 views

What is the significance of the second derivative of a function? [duplicate]

Basically, I just want to know the significance of the 2nd derivative of a function, or what does it tell us.
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0answers
29 views

Minimum Time Through Curve

Consider the following conditions: A level road with the path $\vec{r} = \left<x(s),\ y(s)\right>$ A max speed of $v_{max}$ A vehicle with the mass $m$ Tires with a static friction coefficient ...
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1answer
59 views

The theater, the actors and the 'graviton' [duplicate]

In my perception of the universe, there's the theater which is the 'spacetime' and the actors meaning the 'particles'. If i got it right, GR claims that the 'actors' effect the 'theatre' by 'bending' ...
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2answers
174 views

Do electric charges warp spacetime like stress-energy?

I have read these questions: Does charge bend spacetime like mass? Why is spacetime curved by mass but not charge? Where John Rennie says: "Charge does curve spacetime." And where Frederic Thomas ...
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1answer
100 views

Value of the invariant $R_{\mu \nu}F^{\mu \nu}$

Is there a simple way to find the value of $R_{\mu \nu}F^{\mu \nu}$ (where $R_{\mu \nu}$ is the Ricci tensor and $F^{\mu \nu}$ is the electromagnetic tensor), knowing that it is an invariant? ...
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1answer
102 views

What is the physical significance of Gaussian curvature in condensed matter physics?

In basic models concerning two-level systems, we deal with manifolds such as the Bloch sphere and torus. I believe that the Chern number is what dominates the theory in terms of ties to differential ...
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1answer
132 views

Space-time curvature VS light diffraction

Thanks to YouTube suggestions I saw some videos. One is about the bright spot in the middle of a coin shadow, which is called the Poisson spot, that is related to light diffraction. (https://youtu....
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2answers
265 views

Trace of the Riemann Curvature Tensor

Referring to Wald's General Relativity, I have two questions. Let ${R_{abc}}^d$ be the Riemann curvature tensor. The author has never defined what it means by "trace of a tensor" before page 40 of ...
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How can energy density (of the same sign) have opposite effects on the curvature of the universe?

Actually I have been inspired by the post Why does dark energy produce positive space-time curvature? to ask the following question. In difference to the just cited post I will take the pressure out ...
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1answer
98 views

Why doesn't gravity make the universe have positive curvature?

If mass/energy causes positive curvature in spacetime, and measurements of the curvature of the universe show that it is flat (or very close to flat), then is there something causing negative ...
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0answers
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Anomalous curvature coupling corrections for $Dp$-branes worldvolume actions

The Chern-Simons term of an (abelian brane) is commonly written as $$ \sim\int_{\mathcal M_{p+1}}\sum_iC_{i}[e^{2\pi\alpha'F+B}], $$ where $C_i$ is the background Ramond-Ramond $i$-form, $F$ is the ...
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1answer
58 views

Relation between $r$ and $R$ [closed]

Reference: here Given 3-dim Euclidean metric in spherical coordinates by $$ds^2 = dr^2 + r^2 d\theta^2 + r^2 \text{sin}^2 \theta d\phi^2 \tag{1}$$ so restricting to $$r=R=const. \tag{2}$$ where $...
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1answer
195 views

Prove that deSitter-Schwarzschild metric with constant Ricci scalar is Einstein

Reference: this paper page 4 Prove that the following metric is Einstein Given deSitter-Schwarzschild metric with mass $m > 0$ and scalar Ricci $R=2$ by \begin{align} \bigg( 1-\frac{r^2}{3}-\...
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2answers
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Bubble with varying pressure over surface

I was wondering if it's possible to have a soap bubble with a varying pressure over its surface? That is, such a bubble would not have constant internal pressure at all points. I am specifically ...
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3answers
151 views

What does the metric condition $\nabla_\rho g_{\mu\nu}=0$ in General Relativity intuitively mean for an observer measuring distances?

In General Relativity, the following condition hold: $\nabla_\rho g_{\mu\nu}=0$, where $g_{\mu\nu}$ is the metric of spacetime which has to do with measuring distances and angles and $\nabla$ is the ...
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1answer
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Does the space and time wraping depends on the relative velocity of frame of reference? [closed]

why i am asking this question is that, i thought that photon takes no time to reach the earth or any other place in the universe due to length contraction. the reason for this is space wraps completly ...
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2answers
222 views

If space time can ripple can it shear/tear/shatter? [duplicate]

If gravitational waves role through space time, gravity distorts it etc what would it take to 'tear'it? Simply a black hole or other point singularity? Or does that even do it? Any amount of energy ...
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1answer
129 views

Misner, Thorne and Wheeler, Box 9.2 Commutator … doesn't make sense to me

I apologize for the goofy commutator $\left[\left[\_,\_\right]\right]$ notation. MathJax doesn't like my \llbracket \rrbracket notation. And I religiously use $\left[\dots\right]$ for function ...
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1answer
328 views

Feynman’s “hot plate” versus embedding diagrams?

I just read this lecture by Feynman, where he describes curved two-D space as a “hot plate” where the temperature varies from place to place and it causes the yardsticks to change length and even bend....
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1answer
258 views

Is it possible to bend space 'upwards'? [duplicate]

According to the theory of general relativity 'space' can be bend like a fabric. Objects with a lot of mass bend spacetime like a well or a bowling ball on a stretched blanket. It (images) always ...
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1answer
107 views

Is spacetime in an ellipse around a massive object, or does it just slope down towards the massive object?

In some discussions, people seem to imply that spacetime is in an actual ellipse around a massive object so that, for example, the planet orbiting a star is actually traveling in a straight line in ...
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1answer
184 views

Null Riemann tensor implies no gravity proof?

It's easy to show that no gravity implies a null Riemann tensor, but how could I prove that a null R. tensor implies no gravity?
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2answers
184 views

Doubt-Schild's argument about spacetime curvature

Reading MTW Gravitation book, to me is unclear the point related to the Schild's argument about spacetime curvature. It is basically the following: Consider two observers at rest in the ...
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1answer
1k views

Why is Minkowski spacetime in polar coordinates treated in texts as flat spacetime?

Taking 3-D Minkowski spacetime line element in General Relativity: $$ds^2=-c^2dt^2+dx^2+dy^2+dz^2, $$ when considering a change into spherical coordinates leads to: $$ds^2=-c^2dt^2+dr^2+r^2\left(d\...
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6answers
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Does gravity CAUSE the bending of spacetime, or IS gravity the bending of spacetime? [closed]

In reading these discussions I often see these two different definitions assumed. Yet they are very different. Which is correct: Does gravity CAUSE the bending of spacetime, or IS gravity the ...
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1answer
59 views

Inertial and gravitational Mass

Why definition of mass is not stated as " the property of object to change radius of curvature of space time fabric is called mass"
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1answer
56 views

Example of a Riemannian metric in $(1+1)$D by oscillating geodesics

Newtonian gravitational potential for one mass gives oscillating motions to test particles. Can a somewhat similar movement be produced by ($2$-dimensional $+-$) curvature? I mean an explicit metric ...
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1answer
258 views

Relation between curvature and metric determinant

This is my first question so please let me know if I'm asking my question appropriately! I've been trying to build up an intuition for what curvature of spacetime represents physically. Mostly I have ...
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1answer
318 views

If the space-time curve was quantified and has a mathematical function what would the derivative of the function mean? [closed]

Say I have a massive object: This object as we know causes spacetime to bend and curve. The "maximum curve" which I would define as the line in space time that runs directly through the center of ...
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3answers
153 views

Do shapes of matter affect space-time distortion?

We know heavy objects bend space-time, but does the curvature only depends on the mass? Or different shapes bend space time differently?
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1answer
106 views

Can image method (method of image charges) of electrostatics be used for curved surface?

Suppose we have a fixed point charge in front of a half cylinder. Can we find energy of system using this method? If so, how? I have seen image method being used for infinite sheets. Can we use it if ...
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1answer
111 views

Is it accurate to say space moves because spacetime is curved?

I've learned from PBS Space Time videos that space rushes into a black hole's event horizon at the speed of light. I'm trying to square this with the analogy of stretched rubber sheet everyone uses; ...
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1answer
98 views

Equivalence principle and bending of a light ray

In some explanations of a phenomena of curved spacetime we have an example of a light ray emitted in a freely falling space ship perpendicular to the direction of a free fall. In this case, for an ...
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2answers
100 views

How is time affected by the bending of space-time?

By Newton's Laws, we can calculate the effect of gravity, but he didn't give the actual reason for gravity. I searched the internet for it, but all the answers have this "bending of space-time" stuff. ...
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1answer
301 views

How to derive the formula for the second covariant derivative?

If the first covariant derivative is derived by taking the partial derivative of a vector and applying the product rule to the vector components and the vector bases, couldn't the same thing be done ...
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3answers
113 views

What generates the curvature which is necessary for curling-up extra dimensions?

To say it right away, I am not an expert in string theory, but I know well General Relativity. So I wonder how the curling up of extra-dimensions which is assumed in many "Kaluza-Klein" like theories (...
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1answer
30 views

If an object is heated, does the curvature of the object at a point effect how heat flows out at that point?

For example: while expediting, I have to move little football/oval shaped bowls containing food. I want to hold it in such a way that, when moving it from station 1 to station 2, I have the least ...
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1answer
139 views

How to explain curvature of spacetime at the time of big bang [duplicate]

According to Einstein mass causes curvature of spacetime ,so he basically defined mass using space i.e (mass is something which causes curvature of spacetime).Now at the time of bigbang there was no ...
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1answer
82 views

A simple way to measure the shape of warped spacetime

Background I've wondered for years about ways to visualize how mass (and hence gravity) affects the curvature of spacetime. The 'bowling ball on a rubber sheet' while being useful for the simplest ...
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0answers
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Visualising/getting inuition for, the manifold resulting from these geodesics?

This is related to another post of mine. I know that the geodesics for flat Euclidean Space are straight lines. But I want to take these curved geodesics and tried to work backwards to determine the ...
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0answers
47 views

Discontinuous changes in manifold's curvature?

First of all, there exists a question on PSE which does seem to pertain to my question below, but not exactly. This is one of those questions which requires perhaps an intuitive rather than ...
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0answers
63 views

Derivative of the quadratic invariant term

I know that the derivative of the contraction of two Ricci tensors with respect to the Riemann tensor must be \begin{equation} \left(\frac{\partial \left(R^{\alpha\beta}R_{\alpha\beta}\right)}{\...