# What is Curvature: Definition and 906 Discussions

In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number.
For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or manifold. This leads to the concepts of maximal curvature, minimal curvature, and mean curvature.
For Riemannian manifolds (of dimension at least two) that are not necessarily embedded in a Euclidean space, one can define the curvature intrinsically, that is without referring to an external space. See Curvature of Riemannian manifolds for the definition, which is done in terms of lengths of curves traced on the manifold, and expressed, using linear algebra, by the Riemann curvature tensor.

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1. ### I Flat universe density parameter

For a flat universe, density parameter Ωuniverse=1. How does negative signage of a constituent density parameter, such as that of curvature index Ωk, which can be 0,1,-1 affect the signage of Ωuniverse? If Ωk were to be converted to its energy density, which is much less than the energy density...
2. ### I What is Earth's escape velocity?

Since space is curved within the Earth's gravitational field, every body that moves there will follow the curvature of space no matter what speed it has, so what will its trajectory be, how will it be straight, only if the launch is made absolutely vertically towards sea level? Only then can it...
3. ### I Intuition regarding Riemann curvature tensor

The Riemann curvature tensor contains second derivatives of metric and squares of the first derivatives. The second derivatives have to be there because they are the ones that cannot be eliminated locally by a choice of coordinates. But other than being a mathematical artifact, is there a...
4. ### Unit tangent vector and curvature with arc length parameterization

(a) $$\frac{ds}{dt}=|r'(t)|$$ $$=\sqrt{(x(t))^2+(y(t))^2+(z(t))^2}$$ $$=\frac{2}{9}+\frac{7}{6}t^4$$ $$s=\int_0^t |r'(a)|da=\frac{2}{9}t+\frac{7}{30}t^5$$ Then I think I need to rearrange the equation so ##t## is the subject, but how? Thanks Edit: wait, I realize my mistake. Let me redo
5. ### I Ricci Scalar For Astronomical Body

What would be a rough estimate for the Ricci scalar curvature of an astronomical object like the sun? Assuming the sun is a perfect fluid and you are calculating the rest frame of the sun, only the density component would be factored in. Assuming the sun is roughly 2*1030 kg. Please just make...
6. ### I Connection b/w curvature and gravitational field-removing coordinate transformations

I'm studying Susskind's GR TTM book, in which he gives a nice explanation of why differential geometry is needed for GR. But there is one gap that I want to fill. The argument is: through a thought experiment, it seems that a uniform gravitation field can be seen as an artifact of going from an...
7. ### Calculating the components of the Ricci tensor

(I) Using the relevant equation I find this to be ## \frac{e^{x}}{2} ##. (II) Using the relation for the Ricci tensor, I find that the only non-zero components are...
8. ### I Force carrier particle trajectories and warped spacetime

Do force carriers follow the curvature of spacetime, or do they travel in perfectly straight lines? With black holes, gravity of course exists, so I'm thinking the force carriers (at least gravitons) don't follow spacetime curvature, since they would never escape the event horizon. Sounds like...
9. ### Interference of 2 spherical waves

I believe that to find the curvature C2 is through the ABCD matrices and that C1 has only one phase inversion compared to C. In addition, that the pattern formed is like Newton's rings but I don't know how to find the sizes of the newton's rings depending on lens parameters
10. ### B Faster Than Light Travel: Exploring the Possibilities of Spacetime Curvature

Based on the current understanding of general relativity, it is possible that curving spacetime in the back of a spacecraft would allow for faster-than-light travel. In general relativity, the curvature of spacetime is determined by the universe's distribution of matter and energy. If a...
11. ### Beam deflection and curvature radius formula doubts

Hi, I am working with leaf springs and studying the derivation of the formula for the deflection of such a structure. The derivation is shown here: My only doubt is how to obtain the following formula: $$\delta=\frac{L^{2}}{8R}$$ where: ##\delta## - deflection, ##L## - length of the beam...
12. ### I Research on conservation of spacetime curvature

After trying to kinda get a picture of the field of play in quantum physics according to the standard model, a question came up. I tried to formulate the known bosons each as a particle transferring some property. 1. Photons transfer electric charge: the electromagnetic force gives attraction...
13. ### MTW Exercise 12.6: Formulating Newtonian Gravity Using Curvature

Hello, I am attempting to work through problem 12.6 in MTW which involves formulating Newtonian Gravity using Curvature as opposed to the standard formulation. This is a precursor before standard GR. In it he states that the curvature tensor in this formulation is as follows...
14. ### I Density parameter and curvature index

For Ω=1, κ=0. Does the value of κ simply follow from the value of Ω, or can its value have an independent existence? So if Ω>1, does κ have to be 1?
15. ### I The curvature and size of the Universe

According to the Planck 2018 results, the curvature component of the density parameter of the universe is Ωκ=0.001±0.002. From this data, would it be possible to determine the greatest possible positive curvature of the universe and the radius of the corresponding 3-sphere?
16. ### I Reconciling 2 expressions for Riemann curvature tensor

I'm reading Carroll's GR notes and I'm having trouble deciphering a particular expression for the Riemann curvature tensor. The coordinate-free definition is (eq. 3.71 in the notes): $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ An index-based expression is also given in (eq...
17. ### I Calculate Gaussian Curvature from 4D Metric Tensor

I've been trying to find a way to calculate Gaussian curvature from a 4D metric tensor. I found a program that does this in Mathematica using the Brioschi formula. However, this only seems to work for a 2D metric or formula (I would need to use something with more dimensions). I've found...
18. ### I have a question about the mathematical model of a reflective curve

Hello : Have a question regarding the mathematical model of reflective curve where could i find information on it ? (pdf , webpages , ebooks ,...etc ) Other than Wikipedia Best Regards HB
19. ### I The curvature of space and curvature of spacetime

Hi, The quote below has been taken from this article, https://math.ucr.edu/home/baez/einstein/node2.html, which I came across. The quote doesn't make any sense to me, especially the part in boldface. Could you please help me with it?
20. ### A Curvature & Connection Without Metric

In the absence of a metric, we can not raise and lower indices at will. There are two sorts of Christoffel symbols, Christoffels of the first kind, ##\Gamma^a{}_{bc}## in component notation, and Christoffel symbols of the second kind, ##\Gamma_{abc}##. What's the relationship between the two...
21. ### I Spacetime Curvature via Triangle

I understand the mechanism of defining the curvature of a 2D manifold via triangle. But I don't understand how this works in 3D. Meanwhile, Lawrence Krauss mentioned in his book A Universe from Nothing it does. How does this work in 3D?
22. ### A Anti-self Dual Part (2,2) Riemann Curvature Tensor

i am facing problems in the definition of dual oF some objects which has pair of anti symmetric indices e.g. Weyl curvature tensor. Double dual is there in the literature but given that how to find the anti self dual part of that. the problem is written in attached the file.
23. ### I Diverging Gaussian curvature and (non) simply connected regions

Hi there! I have a few related questions on Gaussian curvature (K) of surfaces and simply connected regions: Suppose that K approaches infinity in the neighborhood of a point (x1,x2) . Is there any relationship between the diverging points of K and (non) simply connected regions? If K diverges...
24. ### MTW Ex 21.15 - Curvature independent of Lapse and Shift functions

Can anyone out there give me a hint as to where to start with this problem? I've been looking at it for a while and can't see a way forward. What exactly is "the curvature itself" here?BTW I think the dynamic initial value equations 21.116 and 21.117 are incorrect. MTW should have inserted to...
25. ### I Calculate Ricci Scalar & Cosm. Const of AdS-Schwarzschild Metric in d-Dimensions

I know some basic GR and encountered the Schwarzschild metric as well as the Riemann tensor. It is known that for maximally symmetric spaces there is a corresponding Riemann tensor and thus Ricci scalar. Question. How do you calculate the Ricci scalar ##R## and cosmological constant ##\Lambda##...
26. ### A General Relativity: Section 5.1 Homogeneity & Isotropy Analysis

From the section[5.1] of 'Homogeneity and Isotropy' from General Relativity by Robert M. Wald (pages 91-92, edition 1984) whatever I have understood is that - ##\Sigma_t## is a spacelike hypersurface for some fixed time ##t##. The hypersurface is homogeneous. The metric of whole space is ##g##...
27. ### B What is the relationship between expansion and curvature?

Physics novice here reading pop sci cosmology. Please bear with me. Premise 1: Whether or not expansion is slowing down or speeding up depends on a battle between two phenomena: the attractive gravitational pull of matter and the repulsive gravitational push of dark energy. What counts in this...
28. ### A Sign of Curvature in Flamm's Paraboloid: Negative?

A question of sign. Is the curvature of Flamm's paraboloid positive or negative? If I've gotten the signs correct, it's a negative curvature. This is the opposite of the positive curvature of a sphere, and it implies that that geodesics drawn on Flamm's parabaloid should diverge. I think...
29. ### B Spatial curvature around a spherically symmetric mass

Suppose I measure the circumference of a circular orbit round a massive object and find it to be c. Suppose I then move to a slightly higher orbit an extra radial distance δr as measured locally. If space was flat I would expect the new circumference to be c + 2πδr. Will the actual measurement...
30. ### I Curvature of 3D Graph on Point w/ Directional Vector

I know curvature (k) of a 2 dimensional graph y(x) is equal to y''/(1+(y')^2)^(3/2), were y' is the first derivative of y with respect to x, and y'' is the second derivative of y with respect to x. Is there a formula for the curvature at a point on a 3 dimensional graph z(x,y)? The curvature...
31. ### I Global coordinate chart on a 2-sphere

Hi, I know there is actually no way to set up a global coordinate chart on a 2-sphere (i.e. we cannot find a family of 2-parameter curves on a 2-sphere such that two nearby points on it have nearby coordinate values on ##\mathbb R^2## and the mapping is one-to-one). So, from a formal...
32. ### I Curvature Index in Flat Universe with Cosmological Constant?

Is the curvature index κ necessarily zero in a flat universe with cosmological constant?
33. ### I Spacetime Curvature: Exploring Eddington's Experiment

Here's what I read in a closed thread asking about the meaning of spacetime curvature: So does that mean if we were able to move to some spot "way above the earth/sun/distant galaxy" where we could watch light and trace it from a distant galaxy (which is otherwise hidden from us) pass around...
34. ### MHB Computing Radius of Curvature

How did the author compute the highlighted term 2 from the highlighted term 1 in the following answer to the given question? If $\rho =\frac{d\psi}{ds}$, then the term 2 should be $\upsilon^2 \frac{d\hat{T}}{d\psi}\rho$, but instead, it was written...
35. ### I Discussing Interior Schwarzschild Proper Lengths & Gaussian Curvature

I'd love have a little discussion about the Interior Schwarzschild Solution. Here's a diagram I slapped together to illustrate the key points. (I assume everyone reading this familiar with embedding diagrams, and using an axis to 'project' a value, in this case the spatial z-axis is replaced by...
36. ### I Negative curvature in low dimensions

It's probably the hyperbola, but I don't see how it's curvature is negative. It looks like 2 parabolas. thanx, paige turner
37. ### Proving ##C## is constant in 4-dim ##R_{\mu\nu}=Cg_{\mu\nu}##

This question wasn't particularly hard, so I assume metric compatibility and input Ricci tensor to the left side of Einstein's equation. $$R_{\mu\nu}-\frac{1}{2} Rg_{\mu\nu}=Cg_{\mu\nu}-\frac{1}{2} (4C)g_{\mu\nu}=-Cg_{\mu\nu}$$ Then apply covariant derivative on both side...
38. ### I Riemann Curvature: Understanding Parallel Transport on 1D Rings

Everyone who is currently studying GR must be familiar with this picture. We find Riemann curvature by paraller transport a "test vector" around and see whether the vector changes its direction. My question. How does it work with one dimensional Ring? A geomteric ring is intuitively curved but...
39. ### I Are Tidal Forces Curvature of Space?

I've heard it and I've read* it before, so I just want to make sure I understand this so I never have to wonder about it again. So, are tidal forces exactly curvature of space? Here's why I think the answer to that is yes: .I've seen a spacetime interval equation which has a coefficient on...
40. ### Riemann curvature coefficients using Cartan structure equation

To calculate the Riemann coefficient for a metric ##g##, one can employ the second Cartan's structure equation: $$\frac{1}{2} \Omega_{ab} (\theta^a \wedge \theta^b) = -\frac{1}{4} R_{ijkl} (dx^i \wedge dx^j)(dx^k \wedge dx^l)$$ and using the tetrad formalism to compute the coefficients of the...
41. ### I Spacetime and spatial curvature

Why do we call it spacetime curvature of gravitation and spatial curvature of the universe? Why don't we call it spacetime curvature of the universe?
42. ### B Simple reasoning that the equivalence principle suggests curvature

I am a high-school teacher and a PhD. student. I am looking for ways to introduce my students to GR. In my treatment, I speak about the equivalence principle and later about curvature in general and consequently that of spacetime. I am missing a connection of these two parts that would be...
43. ### I Changing curvature as a universe evolves

Hi again. I'm still off work and struggling to learn some physics. I'm searching for discussion about the possibility of a universe changing its curvature as it evolves. I'm still new here and so I'd be grateful for advice either about: (i) How to search for past discussions, or...
44. ### Find the osculating plane and the curvature

I know the osculating plane is normal to the binormal vector ##B(t)=(a,b,c)##. And since the point on which I am supposed to find the osculating plane is not given, I'm trying to find the osculating plane at an arbitrary point ##P(x_0,y_0,z_0)##. So, if ##R(x,y,z)## is a point on the plane, the...
45. T

### A On the different ways of determining curvature on manifolds

Hello. Why do we have different ways of determining curvature on manifolds like the sectional curvature, the scalar curvature, the Riemann curvature tensor , the Ricci curvature? What are their different uses on manifolds? Do they allow each of them different applications on manifolds? Thank you.
46. ### B Explaining Spaghettification w/ Spacetime Curvature

I enjoy explaining spacetime curvature to people with a rank-beginner understanding of GR. But someone asked about that favorite concept in pop-sci, spaghettification. I'm having a hard time with it. If you fell into a black hole, there's no reference frame within which you could describe...
47. ### Exercise 16, chapter 3 (Tetrad) in Carroll

My attempt at solution: in tetrad formalism: $$ds^2=e^1e^1+e^2e^2+e^3e^3≡e^ae^a$$ so we can read vielbeins as following: \begin{align} e^1 &=d \psi;\\ e^2 &= \sin \psi \, d\theta;\\ e^3 &= \sin⁡ \psi \,\sin⁡ \theta \, d\phi \end{align} componets of spin connection could be written by using...
48. ### I Radius Excess of Curvature in Relativity: Explained

I think the best place to put this post is the section on special and general relativity. Reading Feynman’s lecture n.42 , volume II here linked : https://www.feynmanlectures.caltech.edu/II_42.html I’ve met the following formula 42.3 for the radius excess of curvature, that Feynman attributes...
49. ### I Riemann Curvature Tensor on 2D Sphere: Surprising Results

I have worked out (and then verified against some sources) that ##R^\theta_{\phi\theta\phi} = sin^2(\theta)##. The rest of the components are either zero or the same as ##R^\theta_{\phi\theta\phi} ## some with the sign flipped. I was surprised at this, because it implies that the curvature...
50. ### I Const Curvature Scalar & 3-Torus: Is It Maximally Symmetric?

Spatial slices of the Robertson-Walker metrics are maximally symmetric so they must have a constant curvature. Is it true that in three Riemannian dimensions that a constant curvature scalar determines whether the volume is finite or infinite? Carroll seems to have given a counter-example for...