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.

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  1. T

    Radius of curvature of 1 MeV electron in 1 Tesla B field

    i need to calculate radius of curvature of 1 MeV KE electron in 1 Tesla magnetic field. r = mv / eB what is the (relativistic) speed, v, of the electron? (ans: 0.941 c ??) then i think you use p(rho) = mo. V / sqrt (1 - v^2 /c^2) = mv (ans: 7.59E-22 kg m/s) e = 1.6E-19 C B = 1...
  2. A

    Physical interpretation of spacetime curvature?

    Folks, I'm in the process of trying to understand spacetime curvature in general relativity. My question might sound odd, but I'm wondering how to best conceptualize spacetime distortions due to a moving mass. If there is a large mass, e.g. a planet, moving through spacetime, the curvature...
  3. C

    How do we argue that gravity is not a force due to curvature of space-time?

    Homework Statement How do we argue that gravity is not a force due to curvature of space-time? Homework Equations I'm new.. I don't even understand the eqn of tensor calculus. The Attempt at a Solution No force is needed for as massive objects follows the curvature of spacetime...
  4. S

    Yau's result for the Ricci curvature on Kahler manifold

    Hi, I've been reading through Yau's proof of the Calabi conjecture (1) and I was quite intrigued by the relation R_{i\bar{j}} = - \frac{\partial^2}{\partial z^i \partial \bar{z}^j } [\log \det (g_{s\bar{t}}) ] derived therein. g_{s \bar{t} } is a Kahler metric on a Kahler manifold (I'm...
  5. O

    Measuring curvature with parallel transport

    Parallel transport, as one means of quantifying the curvature of a coordinate space, enables changes in a vector's components, when it is carried around variously oriented loops in that space, to be properly measured, i.e. by comparisons made at the same location. Those changes which are...
  6. K

    How much curvature are we talking about?

    Homework Statement I've attached a picture which include the question -- I found it on the net, and it's exactly the same as my homework. Homework Equations I'm not sure which formula to use. It's due pretty soon, so please help! :cry:
  7. L

    What is the Curvature of a Helix Given by a Parametric Equation?

    Homework Statement Find the curvature of a helix given by the parametric equation r(t)=<acost, asint, bt> where a and b are real numbers Homework Equations I know k=|T'(t)/r'(t)| The Attempt at a Solution and I believe the answer to be k=b/(a2+b2)1/2, I just don't know how to get there
  8. L

    Finding Curvature of Helix Given Parametric Equation

    Homework Statement Find the curvature of a helix given by the parametric equation r(t)=<acost, asint, bt> where a and b are real numbers Homework Equations I know k=|T'(t)/r'(t)| The Attempt at a Solution and I believe the answer to be k=b/(a2+b2)1/2, I just don't know how to get there
  9. T

    Proof: Curvature Zero -> Motion along a line

    Homework Statement Proof that, if a particle moves along a space curve with curvature 0, then its motion is a along a line.Homework Equations K=\frac{||r'(t)\times r''(t)||}{(||r'(t)||)^3} (curvature of a space curve)The Attempt at a Solution Assume the curve is smooth, so r'(t) cannot be the...
  10. A

    Acceleration and Curvature of space-time

    I'm confused, but when objects travel along the straight lines in curved space-time, do they undergo acceleration? We know that following geodesics is equivalent to inertial motion (one example is free-fall), but when these inertially moving objects travel in curved spacetime, they accelerate...
  11. N

    Curvature of horizon using trig

    Hello! In New Scientist this week (actually next week!), there was a question concerning the curvature at the horizon. The formula is as follows for the distance to the horizon: (2*6373*h)^1/2 km; where h is the height of the individual from the ground. Using the exaple it states a towe...
  12. R

    Curvature forumula of a planar trajectory

    Homework Statement http://steam.cs.ohio.edu/~cmourning/problem1.jpg If the image doesn't load (and it might not, although I'm not sure why), it can be found at: http://steam.cs.ohio.edu/~cmourning/problem1.jpg Homework Equations Part of the problem is I'm not entirely sure what all the...
  13. W

    Find the curvature of the curve

    Homework Statement Given a parametric curve, \alpha(t) = (x(t),y(t) ), not necessarily arc length parameterized show that the curvature is given by: k = \frac{x'y'' - y'x''}{|\alpha'|^{3}} Homework Equations As I understand this the curvature is defined from the point of view of...
  14. J

    Riemann curvature tensor as second derivative of the metric

    It is a standard fact that at any point p in a Riemannian space one can find coordinates such that \left.g_{\mu\nu}\right|_p = \eta_{\mu\nu} and \left.\partial_\lambda g_{\mu\nu}\right|_p. Consider the Taylor expansion of g_{\mu\nu} about p in these coordinates: g_{\mu\nu} = \eta_{\mu\nu}...
  15. Q

    Unraveling the Mysteries of the Riemann Curvature Tensor

    Homework Statement (My first post on this forum) Background: I am teaching myself General Relativity using Dirac's (very thin) 'General Theory of Relativity' (Princeton, 1996). Chapter 11 introduces the (Riemann) curvature tensor (page 20 in my edition). Problem: Dirac lists several...
  16. P

    What is the cause of spacetime curvature?

    hey folks, as far as I understand, according to Einstein's general theory of relativity, any mass that exists in spacetime causes a curvature in it, right?! now, my question is: does this curve take place in the time dimension (ct) or in spacetime (ct,x,y,z) itself?
  17. J

    Determination of Riemann curvature tensor from tidal forces

    Hi, Given a large number of test particles N, it should be possible to determine the Riemann curvature tensor by tracking their motion as they undergo geodesic deviation. Is there a minimum number N that will achieve this in any situation, or does it vary from problem to problem? How...
  18. snoopies622

    Unveiling the Non-Zero Components of Riemann Tensor for Schwarzschild Curvature

    Say, does anyone happen to know the non-zero components of the Riemann (curvature) tensor for the Schwarzschild metric using r,\phi,\theta and t? Thanks.
  19. R

    Negative Curvature: Can We Draw an Infinite Triangle?

    Can we draw an infinite equlateral triangle on a negatively curved surface?
  20. N

    Is the Radius of Curvature of the Universe Constant Over Time?

    Hi all. Is the radius of curvature of a Universe-model constant through-out the Universes' lifetime? Or does this have to be adjusted depending on the redshift we are looking at?
  21. M

    Solving the Curvature Problem for Curve C

    Homework Statement A Curve C is given by the polar equation r=f(theta). Show that the curvature K at the point (r, theta) is K=|2(r')^2 - rr'' + r^2| -------------------- [(r')^2 + r^2]^(3/2) *Represent the curve by r(theta) = r<cos theta, sin theta> Homework Equations I...
  22. R

    Positive Curvature: Maximum Area of Equilateral Triangle

    what's the maximum area of an equilateral triangle that can be drawn on a positive curvature?
  23. I

    Calculate how mass creates a curvature in relativity

    Does exist a system to calculate how mass creates a curvature in relativity, and how it creates redshift?
  24. J

    Gravity and curvature in spacetime

    I'm am new to the forums and have a quick question. When I see diagrams of objects in the universe on the spacetime fabric, the bottom of the object seems to be touching the fabric. My question is, does the fabric touch Earth for example on the south pole like presented in the pictures, or on...
  25. I

    How Relativity say about redshift in terms of curvature?

    How Relativity say about redshift in terms of curvature?
  26. F

    Gaussian Curvature of (x^2+y^2+1)^-2

    1. Homework Statement Is the gaussian curvature at a point on the surface \frac{1}{(x^2+y^2+1)^2}? 2. Homework Equations shape operator: S(\textbf{x})=-D_\textbf{x}\hat{\textbf{n}}=\frac{\partial (n_x, n_y)}{\partial (x,y)} Gaussian Curvature = |S(\textbf{x})|...
  27. F

    Gaussian Curvature: Calculating for g(x,y)=xy\frac{1}{(x^2+y^2+1)^2}

    Homework Statement Is the gaussian curvature at a point on the surface g(x,y)=xy \frac{1}{(x^2+y^2+1)^2}?Homework Equations shape operator: S(\textbf{x})=-D_\textbf{x}\hat{\textbf{n}}=\frac{\partial (n_x, n_y)}{\partial (x,y)} Gaussian Curvature = |S(\textbf{x})| \hat{\textbf{n}}=\frac{\nabla...
  28. M

    Meaning of Curvature Scalar (R) in GR & Its Evolution

    What is the meaning of the curvature scalar (R) in GR? More precisely, what is the meaning of it's evolution? Why when we are concerning the solar system we take R to be small and when we are concerning the cosmological scales the we assume R to be large? Thanks in advance.
  29. G

    Is Spin a Geometrical Indicator of Elementary Particles' Nature?

    According to contemporary ideas the spin of elementary particle is a certain mysterious inner moment of impulse for which it is impossible a somewhat real physical picture to create. The absence of spin visual picture, in opinion of a number of authors leaves the regrettable gap in quantum...
  30. S

    How can we tell if a given tensor is a curvature tensor?

    Under what circumstances do we know whether a given tensor of 4th rank could be the curvature tensor of a manifold? For instance, if I specify some arbitrary functions R_{ijkl} (with the necessary symmetries under interchange i<->j, k<->l, and ij<->kl), is it necessarily the case that there is a...
  31. maverick280857

    Definition of Curvature for Sphere in Relativistic Cosmology

    Hi. I am reading "An Introduction to Modern Astrophysics" by Carroll and Ostlie, for a summer project. In section 27.3 (Relativistic Cosmology) the curvature of a sphere is given by 6\pi \frac{C_{exp}-C_{meas}}{C_{exp}A_{exp}} The situation is as follows: Consider a sphere of radius...
  32. D

    Lightwave moving towards/upwards curvature

    I don't know why this question puzzles me... I believe I can understand the general idea that a lightwave moving in the vicinity of a source of gravity would be deflected by it, as in the "rubber sheet" model, and would curve slightly towards the mass. But is this also valid when the light...
  33. redtree

    Relativistic mass and space-time curvature

    Does relativistic mass curve space-time, i.e., does relativistic mass affect the gravitational field of an object?
  34. D

    Radius of Curvature of particle

    A particle is fired into the air with an initial velocity of 60m/s at an angle of 54 degrees from the ground. At time t=5.443, what is the radius of curvature of the path traveled by the particle? I started by coming up with a vector equation for the path traveled by the particle, using the...
  35. L

    Dual Gauge Curvature for U(1) and SU(3)

    http://camoo.freeshell.org/25.8.pdf" Laura Latex source below. I won't be changing this if I edit the file, it's just for convenience if you want to grab latex code. n sec. 25.8, he says "recall the dual $^\ast F$ of the Maxwell tensor F. We could imagine a 'dual' U(1) gauge...
  36. L

    Dual of Maxwell tensor as gauge curvature

    please see http://camoo.freeshell.org/dual_gauge.pdf" Thanks Laura
  37. R

    Tangent and normal acceleration, curvature radius

    What exactly are the tangent and the normal accelerations of a projectile motion and how are they expressed mathematically? What is curvature radius? What is its expression? How is it derived ?
  38. J

    Magnetic field curvature question

    Homework Statement What is the minimum radius of curvature for an alpha particle, He, moving at 2.0x10^-6m/s in a magnetic field of 2.9x10^-5T? Homework Equations F=qvBsin(theta)...then F=BILsin(theta) The Attempt at a Solution F=qvBsin(theta)...
  39. J

    Curvature of space; curvaure of spacetime

    I have a question about spatial curvature. Before asking the question, I will summarize my understanding of the distinction between spatial curvature and spacetime curvature. If I have this wrong, I would appreciate corrections. Spacetime curvature occurs locally in the presence of gravity...
  40. M

    Radius of Curvature: Definition & Meaning

    What is radius of curvature and why is it equal to 2f where f is the focal length of a lens or a mirror ?
  41. X

    Understanding the Curvature of Time: Exploring Perception and Experience

    We can all see what curvature of space looks like, just by throwing a ball and watching it follow the natural geodesic. But what does curvature of time look like? How do we experience it? We typically experience the passage of time in what seems to be a forward linear manner. The...
  42. X

    Does Clock Speed Vary with Gravitational Potential Inside a Spherical Cavity?

    Inside a spherical cavity centered at the Earth's center, is the space-time curvature is 0 or not 0? Would the clock run more slowly?
  43. Z

    Equation of a Curve in R3 with Constant Inclination: Need Help!

    I'm trying to find the equation of a curve in R3 where k=t=a/(s^2+b) where k is the curvature, t is the torsion and a,b are constants contained in R. I've spent weeks on this problem and at the moment it's driving me nuts since I always seem to end up with an impossible integral. Any help or...
  44. Q

    Curvature of a Sphere and Finding the Area of a Geodesic Circle

    I am attempting to prove the following relation between the curvature K of a sphere of radius R and the area A of a geodesic circle of radius a. K = lim_{a\rightarrow 0} \frac{\pi\cdot a^{2}-A}{a^{4}} \frac{12}{\pi} I'm off by a factor of 4 (i.e. I have 3 in the numerator instead of 12)...
  45. P

    Proof of Lorentz Geom. Not Holding in All Acc. Frames - Example

    obviously the equations of tidal forces and geodesic deviation are very similar to lead one to motivate yourself to explore gravity as not a field but as a curved geometry, Einstein also said that if each accelerated frame is locally an inertial one the euclidean geometry of Lorentz can not...
  46. Helios

    Is space curvature misleading?

    Yes, general relativity is out of my depths for now. Now I've often seen drawings of a gravitational source represented by a dimple ( downward ) on a surface. Yet GR never speaks of a fifth dimension. Nor have I ever seen a dimple upwards that I would suppose would represent repulsive gravity...
  47. M

    Solve Gauss Curvature for Ruled Surface: A(s)+tB(s)

    I'm reviewing for my final and there is a question I can't seem to solve. If anyone could help me with it I would appreciate it very much. A ruled surface has the parameterization of the form: x(s,t) = A(s) + tB(s) where A(s) is unit speed, |B(s)| = 1. Show that: K<or= to 0. So...
  48. O

    Question about curvature of space

    I'm new to physics but very curious about it. I'm 18, and probably will also include physics as my second major in college. A topic that has always bothered me is the curvature of space. If space is curved, due to the planets and stars, then why don't the rays of the sun curve around the...
  49. J

    Superclusters and Voids - same curvature?

    Superclusters and Voids -- same curvature? According to the mainstream 'standard model', is the geometric curvature of space believed to be exactly the same within superclusters as it is in within voids? In other words, does the much higher gravitational density within a supercluster...
  50. S

    Is the correct formula for curvature using the arc length parameter?

    So this book I have (Mathematical Methods for Physics and Engineering, Riley, Hobson, Bence) defines curvature as being: \kappa = \left | \frac{d \hat{\textbf{t}}}{d s} \right | = \left | \frac{d^2 \hat{\textbf{r}}}{d s^2} \right | where t hat is the unit tangent to the curve and r hat...
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