What is Curvature: Definition and 910 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. M

    I Uniform gravity's space-time curvature

    In an ideal world with uniform gravity field, what does its space-time curvature look like? Is it non-zero? If not, how a free particle would be accelerated with the point view of space-time curvature? By uniform gravity field, I mean a gravity field with same value, same direction everywhere...
  2. Bishal Banjara

    How to calculate the Riemann curvature at r=2GM?

    As the coordinate singularity at r=2GM doesn't mean a physical singularity as Riemann curvature tensor is smooth although [g][/rr] metric behaves oddly in the case of Schwarzschild solution. Do somebody tell me an authentic reference how is this value is calculated? I have references writing...
  3. W

    Proving the symmetry property of Riemann curvature tensor

    Homework Statement Hi everyone! Just wondering if there's a way to prove the symmetry property of the Riemann curvature tensor $$ R_{abcd} = R_{cdab}$$ without using the anti-symmetry property $$ R_{abcd} = -R_{bacd} = -R_{abdc} $$? I'm only able to prove it with the anti-symmetry property and...
  4. V

    A How are curvature and field strength exactly the same?

    I am watching these lecture series by Fredric Schuller. [Curvature and torsion on principal bundles - Lec 24 - Frederic Schuller][1] @minute 34:00 In this part he discusses the Lie algebra valued one and two forms on the principal bundle that are pulled back to the base manifold. He shows...
  5. T

    I Calculating the curvature drift in a plasma

    How do i get this equation, $$\frac{db}{dl}=(\hat{B}.∇)\hat{b}$$ This equation is a vector whose direction is towards the centre of the circle which most closely approximates the magnetic field-line at a given point, and whose magnitude is the inverse of the radius of this circle. However I...
  6. C

    I Riemann curvature tensor derivation

    Riemann tensor is defined mathematically like this: ##∇_k∇_jv_i-∇_j∇_kv_i={R^l}_{ijk}v_l## Using covariant derivative formula for covariant tensors and covariant vectors. which are ##∇_av_b=∂_av_b-{Γ^c}_{ab}v_c## ##∇_aT_{bc}=∂_av_{bc}-{Γ^d}_{ac}v_{db}-{Γ^d}_{ab}v_{dc} ##, I got these...
  7. Mr_Phil_Osophy

    B Object interactions in relation to space curvature....

    I'm a complete rookie in this field so please correct me where I go wrong, I just really want a better understanding of this subject. So as far as I am aware, mass causes the space surrounding it to curve or bend. What I want to know is how much does it bend the space? is the bending of space...
  8. Grinkle

    B Curvature at the center of a spherical universe

    If the universe is very large relative to the observable universe and it is spherical, and the observable universe is well away from the outside region of the sphere, more towards the center, is spacetime approaching flat for the observable universe? I always assumed the answer is yes, but then...
  9. V

    Lagrangian for a particle in a bowl with parabolic curvature

    Homework Statement A particle of mass ##m## moves without slipping inside a bowl generated by the paraboloid of revolution ##z=b\rho^2,## where ##b## is a positive constant. Write the Lagrangian and Euler-Lagrange equation for this system. Homework Equations...
  10. FelixLudi

    I The Curvature of the Earth

    (i.e the height you need to be at to look forward) (This isn't in physics subforum because it's more of a geometry question, this is NOT a homework question and it's more of a random thought) Basically, I've been thinking what height would you have to be at, to see 1° of Earth's curvature. So...
  11. C

    A Two Dimensional Ricci curvature

    I want to know if there is some simple metric form for Ricci curvature in dimensions generally. In this paper https://arxiv.org/abs/1402.6334 , formula (5.21), the authers seem had a simple formula for Ricci curvature like this ##R= -\frac{1}{\sqrt{-g}} \partial^\mu \big[\sqrt{-g}(g_{\mu\rho}...
  12. L

    The inverse of spacetime curvature?

    Let's say you can bend a paper...how about bending it upward. a slope I'm saying as we saw spactime in 3d...we all know how it looks..the lines are attracted toward Earth but why doesn't it deflects them and maybe negative mass is linked with it. In other words, someone under the trampoline...
  13. andrewkirk

    I Measuring Space Curvature w/ Swarzschild Coordinate System: Q&A

    I have some questions about the curvature of space (NB not of spacetime) near a planet like Earth. Unambiguously defining space curvature requires choice of a coordinate system, so I choose the Swarzschild system. Here are my questions: Would constant-time hypersurfaces under the Swarzschild...
  14. Buckethead

    B Curvature of space vs. curvature of spacetime

    Regarding curvature of spacetime/space: At some given point in a gravitational field, spacetime is curved at that point and this is a constant. (I'm assuming this is true). Although we can talk about the curvature of spacetime, I never hear anyone talking about the curvature of space. Can...
  15. vibhuav

    I Curvature and Schwarzschild metric

    Many textbooks use the space (spacetime, actually, but for now only space is good enough) around a spherically symmetrical Schwarzschild object to demonstrate curvature of space due to gravity. Let’s consider two shells around such a Schwarzschild object (say a neutron star of 1 solar mass)...
  16. V

    I Proving Effects of Stress-Energy Tensor on Curvature

    Hi everyone. Could you help me to find the way to prove some things? 1)Changing of body velocity or reference frame don't contribute to spacetime curvature 2)On the contrary the change of body mass causes the change of curvature in local spacetime I use the assumption that if we have the same...
  17. N

    I Exploring Einstein's theory about the curvature of space

    Einstein's theory states that curvature of space (created by a celestial body around itself) determines the orbital path of other celestial bodies around it within that curved space by a constant lateral force acting towards the centre upon that revolving body. Then why is that a similar force...
  18. Arman777

    I Friedmann Equation - Positive Curvature

    Friedmann Equation without the cosmological constant can be written as; $$H^2=\frac {8πρG} {3}-\frac {1} {a^2(t)}$$ for ##Ω>1## (which in this case we get a positive curvature universe), and for a matter dominant universe; $$ρ=\frac {1} {a^3(t)}$$ so in the simplest form the Friedmann equation...
  19. D

    I On the Gaussian Curvature of time-like surfaces

    Firstly, I am asking for your patience and understanding because my maths formalism is not going to be rigorous. In another thread here in this forum, I set an example for which now I am asking further instructions. I am going to ask about time-like surfaces immersed in Minkowskian space-time...
  20. Amaterasu21

    I Getting Invariant Curvature from Momentum & Energy

    Hi all, I understand the mathematics behind special relativity pretty well, but I only have a bare conceptual understanding of general relativity. My understanding is that energy, momentum and stress (as described in the energy-stress tensor) are what contribute to space-time curvature and...
  21. tensor0910

    Finding maximum curvature on lnx

    Homework Statement Determine the point on a plane curve f(x) = ln x where the curvature is maximum.[/B]Homework Equations k(x) = || T ' (x) || / || r ' (x ) || k (x) = f '' (x) / [ 1 + ( f '' (x))2 ] 3/2[/B]The Attempt at a Solution f ' (x) = 1/x f " (x) = -1/x2 k(x) = 1/x2 / { [1 +...
  22. dreens

    I Can the Electromagnetic Force be Explained by Curving Spacetime?

    So General Relativity explains the force of gravity as mass/energy induced curvature of spacetime. This correctly predicts gravitational time distortion, nonlinear geodesics and gravitational lensing, the anomalous precession of planetary orbits, the schwarzchild metric, and so on. Could the...
  23. gaaah

    B Gravity Illusion: Questions on Space-Time Curvature Granularity

    I wonder if someone would field a beginner's muse I had: If gravity is just an illusion of the curvature of space caused by mass, does not the matter within that space follow the curve? and what is the granularity of that curvature? Does the curvature exist in the space between the nucleus and...
  24. P

    I Non-zero components of Riemann curvature tensor with Schwarzschild metric

    I was working out the components of the Riemann curvature tensor using the Schwarzschild metric a while back just as an exercise (I’m not a student, and Mathematica is expensive, so I don’t have access to any computing programs that can do it for me, and now that I’m thinking about it, does...
  25. J

    Constant acceleration and radius of curvature

    Homework Statement Homework Equations The Attempt at a Solution The normal acceleration of the particle at any instant is given by an = v2/r . v is the speed at any time and r is the radius of curvature . Minimum radius will occur when ratio v2/an is minimum . I think this will occur when...
  26. F

    I How Does Energy Source Curvature? Exploring the EEP

    TL;DR Why does the Einstein equivalence principle imply that all forms of (non-gravitational) energy source curvature? Now, as understand it, the Einstein equivalence principle (EEP) implies (or at least suggests) that gravity is the manifestation of spacetime curvature, the reason being that...
  27. J

    A On the dependence of the curvature tensor on the metric

    Hello! I was thinking about the Riemann curvature tensor(and the torsion tensor) and the way they are defined and it seems to me that they just need a connection(not Levi-Civita) to be defined. They don't need a metric. So, in reality, we can talk about the Riemann curvature tensor of smooth...
  28. karush

    MHB 13.4.7 Find the curvature of the curve r(t).

    $\textit{$13.4.7$ Find the curvature of the curve $r(t).$}$ \begin{align*}\displaystyle r(t)&=(5+9 \cos 8t) i - (4 + 9 \sin 8t)j + 6k\\ v&= -72\sin(8t)+72\cos(8t)\\ |v|&=\sqrt{(-72\sin(8t))^2 +(72\cos(8t))^2}\\ &=5184\\ \frac{v}{|v|}&=\frac{-72\sin(8t)+72\cos(8t)}{5184}\\...
  29. Ontophobe

    I EM Force vs. Gravity: Modeling Spatio-Temporal Curvature

    Why is it that the EM force can't be modeled on spatio-temporal curvature the way gravity can be?
  30. D

    I What properties allow me to see past Earth's curvature?

    I recently was able to view a 193 foot building from 24 miles away. The base of the building is approximately 15 feet above sea level and my eye level was approximately 9 feet above sea level. I was viewing the building across a Lake. I could see a substantial amount of the building, which...
  31. G

    I Curvature of space and spacetime

    General relativity suggests that path of light is curved around sun. This curvature is not dependent upon frequency of the photon. What is the physical difference between 'curvature of space' and 'curvature of space-time' ? We can make measurements at two points in space at same time. But there...
  32. B

    B GR Curvature at Light Cone Surface: Smooth, Bent, Blocked?

    For example, the curvature due to a mass; does that curvature continue passing from within to outside the mass's light cone? If so, is the mass subject to the external curvature? If not, does the curvature have a discontinuity at the light cone surface?
  33. K

    I The spacetime curvature changed by an object

    Does the amount by which an object changes the spacetime curvature depend on relativistic mass or the rest mass? Through this question I just want to answer whether momentum equals [relativistic mass * velocity] or is it [rest mass * gamma * velocity]. Both the formulas might be the same but I...
  34. DoobleD

    I Influence of Dark Energy and curvature in photon-baryon fluid?

    When I read explanations about the early Universe and the oscillations of the photon-baryon fluid before recombination, effects of the cosmological constant and of the curvature of the Universe on the fluid are never discussed. Only dark matter, baryons, and photons are mentionned. Dark energy...
  35. deuce123

    I Explaining Extreme Spacetime Curvature & Time Physics

    Can someone please explain to me why time drastically slows for anyone near an extremely curves spacetime. I see it as the flow of time almost becomes slowed due to extreme curvature, what can explain this? What do physicists "see" time as? Also, I'm not entirely educated on this topic but I...
  36. FallenApple

    I Can spacetime curvature be transformed away?

    This is probably a bad question, but can it be transformed away? Say Alice is on Earth and Bob is far away in outer space. Bob would think that Alice's clock is running slow. Alice would think Bob's clock is running fast. A third observer, say Carl, anywhere in spacetime would have to observe...
  37. A

    A Gravity as space-time curvature?

    Hi there. I was wondering that if mini or micro black holes could theoretically exist, and if not all black holes need to "devour' matter, then could it be possible that all things we perceive to have gravity could possibly be caused by a mini or micro black hole at the center of massive objects...
  38. J

    B How to do the calculations showing the Universe is flat?

    I've been trying to understand how we know that the observable universe is flat, and I'm having difficulty finding any sources that explain exactly how the calculations were done. On this WMAP website (https://map.gsfc.nasa.gov/mission/sgoals_parameters_geom.html), it says: "A central feature of...
  39. C

    I Extrinsic Curvature Formulas in General Relativity: Are They Equivalent?

    I know two kinds formulas to calculate extrinsic curvature. But I found they do not match. One is from "Calculus: An Intuitive and Physical Approach"##K=\frac{d\phi}{ds}## where ##Δ\phi## is the change in direction and ##Δs## is the change in length. For parametric form curve ##(x(t),y(t))##...
  40. C

    A How to calculate the extrinsic curvature of boundary of AdS_2

    I have a simple but technical problem: How to calculate the extrinsic curvature of boundary of AdS_2? I am not very familiar with this kind of calculation. The boundary of AdS2metric $$ds^2=\frac{dt^2+dz^2}{z^2}$$ is given by (t(u),z(u)). The induced metric on the boundary is...
  41. FallenApple

    A Analyzing Curvature with Confidence Bands?

    So let's say that there are several different treatments for being underweight. One way to deal with this is to make repeated measurements of each patient's weight across time. So I can fit a model to the panel data, regressing weight on treatment as an indicator variable, and then plot the...
  42. Deepak K Kapur

    B Question about Curvature caused by Mass

    A body is moving in deep space substantially away from others influences of gravitational force ( just assume). Suddenly, a big planet appears in It's vicinity. As per relativity it's not some force issuing forth from the planet that attracts the body towards itself. The body just keeps on...
  43. Deepak K Kapur

    B Mechanism of Curvature of Space: Does it Happen Instantly?

    Hi all, How does matter curve space (what's the mechanism)? Does this Curvature happen instantaneously or does it happen at light speed? Thanks in advance.
  44. PeterDonis

    A Does gravitational time dilation imply spacetime curvature?

    In a recent thread, the question came up of whether the presence of gravitational time dilation implies spacetime curvature. My answer in that thread was no: This was based on the obvious counterexample of observers at rest in Rindler coordinates in flat Minkowski spacetime; two observers at...
  45. P

    I Einstein's elevator: gravity without curvature?

    This is a rather old issue, but one that has recently been on my mind. We often say that gravity is the curvature of space-time, with good reason. At the same time, we also talk about the "gravity" in Einstein's elevator, as an example of the equivalence principle. This is also with good...
  46. P

    I Testing Elliptic PDE Solver with Non-Diagonal Metric

    Hello, I am working with numerical relativity and spectral methods. Recently I finished a general elliptic PDE solver using spectral methods, so now I want to do Physics with it. I am interested in solving the lapse equation, which fits into this category of PDEs $$ \nabla^2 \alpha = \alpha...
  47. A

    A Intution behind the definition of extrinsic curvature

    Forgive me for asking a rather silly question, but I have thinking about the following definition of the extrinsic curvature ##\mathcal{K}_{ij}## of a sub-manifold (say, a boundary ##\partial M## of a manifold ##M##): $$\mathcal{K}_{ij} \equiv \frac{1}{2}\mathcal{L}_{n}h_{ij} =...
  48. Peter Martin

    B Einstein's curvature vs. Newton's

    I've read that observations during the solar eclipse of 1919 showed that Einstein's prediction of light's curvature passing near the Sun was exactly twice that of Newton. I have two questions: 1. Why did Newton assume any curvature? Did he think "light particles" had mass? 2. In the physical...
  49. D

    I Use a level to test the Earth's curvature?

    So, I'm sure many of you have seen the experiment recently by a flat-Earth member who brought a level on a plane with him in order to determine whether or not the Earth is flat or round. And yeah, my immediate thought was similarly "how silly!" But it did make me wonder: Could you do it that...
  50. A

    I Understanding Parallel Transport

    I'm currently in a GR class and have come across the notion of parallel transport, and I've searched and searched the last few days to try and understand it but I just can't seem to wrap my head around it, so I'm hoping someone here can clarify for me. The way I picture parallel transport is...
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