What is Curvature of space: Definition and 37 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. V

    A How to derive Quantum Mechanics in curved physical space?

    I am following [this YouTube lecture by Schuller][1] where he finds the appropriate formalism for the quantum mechanics in the physical curved space. Everything makes sense to me but at the very end I see that we find the pull backed connection one-form on the base manifold. He says to the end...
  2. Jaime Rudas

    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?
  3. PainterGuy

    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?
  4. G

    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...
  5. Martian2020

    B General Relativity and the curvature of space: more space or less than flat?

    General relativity. Curvature of spacetime: ok. time dilation: ok. What about space? Curvature is intrinsic and given by complex equations. But could we definitely say is there more space between 2 points along curved space through the star than would be through flat space (no star there) or...
  6. Arman777

    I Curvature of Space in the Context of Cosmology

    Recently I asked a question about the curvature of the universe. https://www.physicsforums.com/threads/constant-curvature-and-about-its-meaning.977841/ In that context I want to ask something else. Is this curvature (##\kappa##) different than the Gaussian Curvature ? Like it seems that we...
  7. 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...
  8. 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...
  9. 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...
  10. 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...
  11. 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...
  12. 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.
  13. Brunolem33

    B Gravity and the curvature of space

    I am not sure I can express clearly my thoughts on this one, but I am going to try. The effect of mass on space, and the resulting gravity, is sometimes described, in a two dimensions graphic, as a bowling ball applying pressure on the surface of a mattress, or something equivalent. Thus, the...
  14. U

    Lifetime of Universe: Limits & Expansion Explained

    I am studying general relativity from Hobson and came across the term 'lifetime' of a closed (k>0) universe, ##t_{lifetime}##. I suppose at late times the curvature dominates and universe starts contracting? Are they simply referring to ##\int_0^{\infty} dt##? If so, would the bottom expression...
  15. xortdsc

    Computing Curvature of Space at Point from Mass - Help Needed

    Hello, given a stationary pointlike particle with mass m at some position, I'm trying to compute just how much space is curved/deformed at a distance r from that particle due to its gravitational field. I'm not really into all that tensor calculus, so I really struggle with the equations given...
  16. T

    Curvature of Space-Time: Why is Covariant Derivative Nonzero?

    I recently watched Susskind video on general relativity. I am unsure why the commutator of the covariant derivative of the vectors is nonzero when there is curvature. E.g. DrDsVm-DsDrVm In flat space, that difference is zero. But why is it non zero in curved space? Someone please enlightened...
  17. C

    Question about Gravity and curvature of space time

    Hello all I just joined this forum so forgive me for jumping right in but I have a question about Gravity and the curvature of space time that I can't get answer with a Google search. My question: though I understand that an object remains in orbit because of the curvature of space time and it...
  18. anorlunda

    Must the curvature of space be constant?

    I was listening to one of Leonard Susskind's cosmology lectures. He talks about the factor K having values of +1/0/-1 corresponding to positive/flat/negative curvature. We don't know what the real value of K is. But then as he discussed K at the big bang and K now, it seemed that he was...
  19. R

    Curvature of Space: What is it?

    Curvature of Space-time: What is it? General relativity talks about curvature of space-time due to mass. What does it actually mean by 'curve'. Is the space made of something that we can say is curving? If space is purely 'empty', then what is getting curve? Or is it that curving is just an...
  20. Q

    Curvature of space and spacetime

    i am trying to understand the relationship between the two on a local and global scale and how these two concepts are related to the Ricci scalar. Is it correct to say that as far as we know on a global scale, spacetime is flat so that the Ricci scalar is zero. If so, what can be said about...
  21. R

    Gravitation as curvature of space vs field theory

    Gravitation is described on one hand as curvature of space in the presence of matter. It is also described as a field acting through gravitons on matter. How can the two views be reconciled?
  22. T

    Is gravity a force, or the curvature of space?

    After some light reading, I'm more confused than ever. Is gravity just a byproduct or effect of the curvature of space? Is it a force that would exist if space didn't curve, even in the presence of mass? (probably a stupid question, sorry!) I've seen various diagrams of the Earth revolving...
  23. L

    Measuring curvature of space around a star

    I am wondering how space geographers would measure curvature of space around a large isolated star. i am thinking of the set up where there are two nearby spheres surrounding the star whose circumferences are already known. The remaining step is to measure the length of a radial geodesic segment...
  24. R

    Gravity and curvature of space

    I could understand gravity involve in object revolving around a big mass as the path of the object is circular. But couldn't understand how gravity is explained when an object thrown up returns back.
  25. D

    The Curvature of Space and Acceleration Confusion

    Like many of these forum dwellers, I've been reading the Elegant Universe and I've hit a fit of confusion. So I've got a couple of questions. In the book, it is explained that accelerated motion results in the warping of space and time (I'm thinking specifically of his example of the rigidly...
  26. M

    Curvature of Space: Explaining Gravitational Forces

    Hi! A google search for my upcomming question lead me to this page, and I am delighted to find an online community of psysics who hopefully will answer my, maybe, very simpel question. In the study of Einstein's General Relativity theory, the picture of a sphere placed in a net representing...
  27. T

    Does compressing a mass to a smaller sphere change the curvature of space?

    If you take a homogenously distributed spherical mass and compress it to a smaller radius while maintaining its overall total energy and momentum, will it change the curvature of space (gravity) outside of the original sphere? According to Newton, it stays the same. However, it seems like...
  28. M

    Curvature of space and dark matter

    Assuming gravity is matter curving space as Einstein says, isn't our theory of dark matter just an assumption that because more gravity is required to explain galaxy formation that it must be caused by unseen matter? Why do we assume that the curvature of space required must be caused by matter...
  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. 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...
  31. 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...
  32. A

    Question on Curvature of Space

    Would it be possible to adjust for the curvature of space between 2 points and so by taking the shortcut (a true straight line) beat a light source in a race between the 2 points whilst traveling at less than light speed?
  33. K

    Dark Energy's effect on the curvature of space time

    Given that Dark energy supplies negative pressure: Does it have mass? Does it curve space? If so, does it help to close the universe? Can something that has negative pressure also make space more closed? Does it affect time dilation? How do you achieve a flat universe when there seems to...
  34. M

    Exploring the Curvature of Space: Tensor Analysis & Beyond

    When Enstein developed tensor analysis because all the other math fell apart, there was another type of math that was developed to measure the curvature of space. What is it? How does it work?
  35. DaveC426913

    Curvature of Space - Is it Still Relevant in Modern Physics?

    I have not been keeping up on current cosmology theory. Once was said that if you flew your "Spaceship of the Mind" X billion light years "thataway", and didn't stop or turn, you eventually would find yourself back where you started. The idea here is that 3-dimensional space is actually...
  36. quasar987

    On the general curvature of space.

    What is actually commonly accepted in the cosmologist society regarding the model of the universe? Do they think C=0 or > 0 or < 0 ? And what makes them think that? Also, how do one get the shapes related to each posibility for C from the equations of GR?