Curvature Definition and 920 Threads

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.

View More On Wikipedia.org
Recent contents Top users
  1. Buckethead

    B Spacetime curvature due to acceleration causing gravity?

    In reviewing one of Einstein's thought experiments, the accelerating elevator in space, and the resulting bending of light passing through the elevator, Einstein's predicted that light will bend in gravity. Now Einstein's original prediction was off by a factor of 2 because he hadn't yet...
  2. R

    Why Centrifugal Force in the Derivation of Curvature Drift

    Homework Statement This is a question about a pretty basic plasma physics derivation. In the standard derivation of the Curvature Drift of a charged particle in a magnetic field with curvature, the force that they use is the "imaginary" centrifugal force (or the force the guiding center sees in...
  3. Arman777

    I Relationship between Energy density and Curvature

    I don't know GR so while answering the question if you prefer not to use that, I would be happy. In the Friedmann Equations, is energy density has an effect on curvature or vice versa? Or they are separate things and they don't affect each other? For example can we have an energy density...
  4. Arman777

    I Why does an Empty Universe have to obey Negative Curvature?

    Its stated that empty universe should have a hyperbolic geometry (Milne Universe) but I don't understand how its possible. $$H^2=\frac {8\pi G\epsilon} {3c^2}-\frac {\kappa c^2} {R^2a^2(t)}$$For an empty universe when we set ##\epsilon=0## we get $$H^2=\frac {-\kappa c^2} {a^2(t)}$$...
  5. H

    I Curvature of space in large regions: zero or not?

    I have read numerous times that the overall curvature of space in extremely large regions -1000s of megaparsecs say - is zero. I also keep reading that the expansion rate of the universe is increasing, and that the universe is resultantly positively curved. I would be interested in a...
  6. A

    B Question regarding if curvature is not constant

    For 2-sphere it is having a curvature of k=1/R ,where R is the radius of the 2-sphere and to make it more generalised we treat the kR as the curvature which is always +1 and is independent of it's radius. My question is how do we treat the curvature term for 3-sphere , And it the curvature term...
  7. A

    B Positive curvature of a 2D surface

    If considering a 2D surface (plane) having polar coordinate r,θ (where r is the distance from the origin and θ is the anticlockwise angle from the base line as usual) The metric is now actually ds2=dr2+r2dθ2 If now this 2D surface is given a positive curvature of +1 (equivalent to the surface of...
  8. Hugh de Launay

    I Does S-T Curvature Change Particle Shape?

    This is an inquiry about some of the details of the bending of the trajectory if a photon by curved space-time (S-T) from the perspective of general relativity (GR). The environment in this scenario contains a black hole at its center, a photon passing by the black hole a km above the event...
  9. P

    A Fundamental definition of extrinsic curvature

    My question is quite simple: what is the fundamental definition of extrinsic curvature of an hypersurface? Let me explain why I have not just copied one definition from the abundant literature. The specific structure on the Lorentzian manifold that I'm considering does not imply that an...
  10. PaulCam

    B Space Curvature & Potential Energy

    So I'm an Software Engineer, not a physicist, nor a mathematician. So I like to work in the qualitative, not the quantitative. Today I hit on a problem. I've been trying to remove the concept of "down" or "inward" from my thinking of gravity and GR. When people show the concept of space/time...
  11. 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...
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. 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...
  17. 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...
  18. 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...
  19. 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...
  20. FelixLudi

    I How High Must You Be to See 1° of Earth's Curvature?

    (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...
  21. 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}...
  22. 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...
  23. 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...
  24. 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...
  25. 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)...
  26. 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...
  27. 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...
  28. 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...
  29. 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...
  30. 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...
  31. 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 +...
  32. 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...
  33. 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...
  34. 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...
  35. 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...
  36. 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...
  37. 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...
  38. 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}\\...
  39. 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?
  40. 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...
  41. 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...
  42. 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?
  43. 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...
  44. 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...
  45. 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...
  46. 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...
  47. 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...
  48. 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...
  49. 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))##...
  50. 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...
Back
Top