What is Curved space: Definition and 80 Discussions

Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Curved spaces play an essential role in general relativity, where gravity is often visualized as curved space. The Friedmann-Lemaître-Robertson-Walker metric is a curved metric which forms the current foundation for the description of the expansion of space and shape of the universe.

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  1. Sciencemaster

    I Gravitational Field Transformations Under Boosted Velocity

    Let's say we have some observer in some curved spacetime, and we have another observer moving relative to them with some velocity ##v## that is a significant fraction of ##c##. How would coordinates in this curved spacetime change between the two reference frames? For example, imagine a...
  2. K

    I Proper time in a curved space

    In special relativity we've the invariant ##\begin{aligned} d s^2=&-d t^2 \\ &+d x^2 \\+d y^2+d z \end{aligned}##. For a clock moving along a worldline the above equation reduces to ##\begin{aligned} d s^2=&-d t^2\end{aligned}## , hence we can say that the time measured by the clock moving...
  3. Ahmed1029

    I What's the definition of angle in a curved space embedded in a higher Eucledian space?

    I don't want to post this in a math forum because it's very basic and I just want a straightforward answer, not something math heavy . What's the definition of angle in a cuved space embedded in a higher eucledian space? Like when I have a spherical surface in 3d eucledian space and want to work...
  4. N

    I Curved space and gravitational waves

    Are gravitational waves purely temporal? An object with no spatial velocity experiences gravity due to temporal velocity?
  5. Y

    A B and A in Curved Space Time: Does \nabla \times A =B?

    By definition of the vector potential we may write \nabla \times A =B at least in flat space. Does this relation hold in curved space? I am particularly interested if we can still write this in a spatially flat Friedmann-Robertson-Walker background with metric ds^2=dt^2-a^2(dx^2+dy^2+dz^2) and...
  6. N

    I Position Vector in Curved Space Time: Explained

    It is said that: It is not possible to write a position vector in a curved space time. What is the reason? How can one describe a general vector in a curved space time? Can you please suggest a good textbook or an article which explains this aspect?
  7. Pouramat

    Energy-Momentum Tensor for Electromagnetism in curved space

    a) I'd separated the Lagrangian into: $$ \mathcal L = \mathcal L_{Max}+\mathcal L_{int} $$ in which ##\mathcal L_{Max} =\frac{-1}{4}\sqrt{-g} F^{\mu \nu}F_{\mu \nu}## and ##\mathcal L_{int} =\sqrt{-g} A_\mu J^\mu## Thus: $$ T^{\mu \nu}_{Max}= F^{\mu...
  8. A

    I Expansion of 3-D positively curved space

    The metric of a 3-D positively curved space is dr2+ Sk(r)2(dθ2+sin2θdΦ2). Now if this space expands with a scale factor a(t) from r to r'. Whether the change in the radial component be a(t)dr and angular component be Sk(r')dθ and Sk(r')sinθdΦ since the change due to expansion is already...
  9. vincent

    I What is it that forces an object down curved space time?

    To explain the concept of curved space time, we often use analogy of rubber sheet. If we put a heavy ball at the centre of sheet then it creates a depression and now a smaller ball will fall towards that heavy ball because of depression. But in this analogy smaller ball is falling down the slope...
  10. P

    A Lense-Thirring effect - General Relativity

    Let us assume a "toy-metric" of the form $$ g=-c^2 \mathrm{d}t^2+\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2-\frac{4GJ}{c^3 r^3} (c \mathrm{d}t) \left( \frac{x\mathrm{d}y-y\mathrm{d}x}{r} \right)$$ where ##J## is the angular-momentum vector of the source. Consider the curve $$ \gamma(\tau)=(x^\mu...
  11. B

    Metric of a globally negatively curved space

    Homework Statement I think I have managed to do the first three parts of this problem ok, but I am struggling with part 4. [/B] A 2D negatively curved surface can be described in 3D Euclidean Cartesian coordinates by the equation: ##x^2 + y^2 + z^2 = −a^2##. 1) Find the 2D line element for...
  12. Chompers

    B Does gravity cause time dilation?

    Hi all, I'm doing some reading about special/general relativity, and have come across the ideas of curved space etc. I've very much a novice in physics, so please excuse my (possibly) stupid questions. For background, I'm interested in writing a sci-fi story, and would like to have at least...
  13. P

    A Understanding Orbital Angular Momentum Coupling to Christoffel Connection

    I am trying to understand Wen and Zee's article on topological quantum numbers of Hall fluid on curved space: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.69.953 They passingly mentiond the fact that a spinning particle with orbital angular momentum $s$ moving on a manifold with...
  14. S

    I Why do planets follow the same curvature at both foci?

    We are told that planets and comets orbit the sun in an ellipse (Kepler's 3 laws) as shown below: We are also told that according to Einstein's theory of gravity, there is no force applied. Implied is that the planets move in straight lines through curved space. We know that the effect of...
  15. felicja

    I What is an angle deficit in the curved space in GR?

    I think this should be equal to the famous precession angle, but with a negative sign: ##d\phi = 6\pi m/r## correct?
  16. mertcan

    I Curved space and curvilinear coordinates

    hi, I really wonder what the difference between curvilinear coordinates in a Euclidean space and embedding a curved space into Euclidean space is ? They resemble to each other for me, so Could you explain or spell out the difference? Thanks in advance...
  17. J

    A Can random walks be applied to String Theory in curved space

    If we study the high temperature limit (near Hagedorn) of a string gas, most of the energy is concentrated in a single long string. If we model the string by a fixed number of rigid links of length ls and calculate the number of possible configurations, we get the density of states: $$\omega(E)...
  18. P

    I If there are gravitons, is space not curved?

    If gravitons are proven to exist, would that mean space is not curved?
  19. naima

    Integration by parts in curved space time

    In this thread, ramparts asked how integration by parts could be used in general relativity. suppose you have ##\int_M (\nabla^a \nabla_a f) g .Vol## Can it be written like ##\int_M (\nabla^a \nabla_a g) f .Vol## plus a boundary integration term (by integrating twice by parts)? I think thay it...
  20. newjerseyrunner

    Angle and trig definitions in curved space

    I was going to ask a question about whether or not pi was constant or changed with curved space. I found the answer on here that it does indeed change. Then I started thinking about the ramifications of that. sine waves are dependent on pi, so they should change too. Does sin(theta) =...
  21. H

    Exploring Physics: A Newbie's Questions on Space and Curvature

    Hello, I just joined. I have no formal background in physics, just curiosity. So, my questions may well be simplistic to most of you. Hopefully, that is permissible, now and then! First question, if space, as it's usually defined, is empty, nothing, how can it be curved? Hankb
  22. P

    Does light curve toward ordered

    If I'm correct in thinking light curves towards objects with less entropy; why wouldn't the path of least resistance be towards entropy?
  23. S

    Hubbard-Stratonovich transformation

    Can one apply the Hubbard-Stratonovich transformation to the exponential of the Laplace–Beltrami operator?
  24. R

    Derivatives of the Lagrangian in curved space

    Follow along at http://star-www.st-and.ac.uk/~hz4/gr/GRlec4+5+6.pdf and go to PDF page 9 or page 44 of the "slides." I'm trying to see how to go from the first to the third line. If we write the free particle Lagrangian and use q^i-dot and q^j-dot as the velocities and metric g_ij, how is it we...
  25. B

    Gravitational Effects of Gyros Near a Massive Object

    Given an object spinning on its own axis, in orbit around a planet with mass and that the object is traveling in the direction of its axis, does the axis continue to point at the same point in infinity as it rotates around the planet or does the axis follow the curvature of space around the...
  26. arpon

    How to define covariant basis in curved space 'intrinsicly'?

    In Euclidean space, we may define covariant basis by the partial derivative of position vector with respect to each coordinates, i.e. ##∂R/(∂z^i )=z_i## But in curved space (such as, the two dimensional space on a sphere) how can we define covariant basis 'intrinsicly'?(as we have no position...
  27. W

    Time dilation and curved space

    I am trying to get an understanding of general relativity one tidbit at a time. I have a vague concept of why curved spacetime causes the effect we call gravity. However, there's an aspect of it (ok, there' are quite many aspects of it, but I'm concentrating on this one right now) that I can't...
  28. craigi

    Metal packing structure and curved space

    In flat space the atoms in a metal have regular packing structures. A slight curvature of space would mean this wasn't geometrically possible. As a consequence do we expect metals to have a significantly lower density with a slight curvature of space? Obviously, this doesn't just apply to...
  29. D

    Why Curved Space Affects Satellite Orbit: The Role of Newtonian Force

    If the space around the Earth is curved according to general theory of relativity no lateral force is required to put the satellite in orbit because when the rocket carrying satellite has reached the certain height the satellite should spontaneously start sliding along the curved path traced...
  30. C

    Regular empirical evidence of curved space or massless photons?

    NOT including the prediction capabilities of the particular math equations of GR or SR. In particular, hard evidence such as, or close to; here's an electron, because we measured it directly, or saw it in an electron microscope. Or here's a cell under a microscope. Or this is a brain scan/MRI...
  31. P

    Dirac equation, curved space time

    Hi when trying to derive this equation, i am stuck on: [\Gamma_{\mu}(x),\gamma^{\nu}(x)]=\frac{\partial \gamma^{\nu}(x)}{\partial x^{\mu}} + \Gamma^{\nu}_{\mu p}\gamma^{p} . This [\Gamma_{\mu}(x) term is the spin connection, if this is an ordinary commutator: a) is it a fermionic so +...
  32. G

    How Do Dirac Spinors Relate to the Ricci Scalar in Curved Spacetime?

    I have to compute the square of the Dirac operator, D=γaeμaDμ , in curved space time (DμΨ=∂μΨ+AabμΣab is the covariant derivative of the spinor field and Σab the Lorentz generators involving gamma matrices). Dirac equation for the massless fermion is γaeμaDμΨ=0. In particular I have to show that...
  33. S

    Parallel transporting of a vector in curved space

    Am I correct in saying that the angular deficit (change in angle) of a vector transported around a closed surface on a curved surface can only be observed by flattening the surface? Actually a further problem- I understand it from the flat sheet to a cone: Cut out a pie from a sheet, Draw a...
  34. F

    Quantum wave equations in curved space

    Hi, I've been looking at the Klein Gordon equation, Maxwell's equation, and the Dirac equation in curved space and I was wondering if there is an underlying formalism regarding how to derive them from their flat space counterparts. What I mean is, at the heart of the whole process for all...
  35. F

    Can the Dirac Delta Function be Applied in Curved Space?

    The Dirac delta function is defined as: \int_{ - \infty }^{ + \infty } {\delta (x - {x_0})dx} = 1 Or more generally the integral is, \int_{ - \infty }^{ + \infty } {\delta (\int_{{x_0}}^x {dx'} )dx} But if the metric varies with x, then the integral becomes, \int_{ - \infty }^{ + \infty }...
  36. Buckethead

    Is curved space cancelled between two massive bodies?

    Considering two equally massive stars stationed at some small distance from each other, would a clock stationed between the two stars equadistance from both tick more slowly due to the proximity to the massive stars, or would the effective cancellation of the gravitational attraction also cancel...
  37. J

    Measure for momentum in curved space

    When I write down a quantum field (for instance to compute T^00 or some expectation value) I write it as an integral over momentum space. If I am working in curved space should this be divided by sqrt [g]? (and why or why not?)
  38. mrspeedybob

    Do Corners Experience Stress in Curved Space?

    Suppose I construct a metal triangle in flat space the sum of the interior angles will be 180°. If I then move the structure into a curved space in which the sum of the interior angles of the triangle will be greater then 180° do the corners of my triangle experience stress? Or, do I simply...
  39. B

    Is curved space allowed to show discontinuities (steps)?

    Recall the gravitational effect of a hollow sphere upon a test mass inside ... no net attraction of the test mass anywhere within the sphere. Let this sphere be large and dense enough to have 1 G of attraction on a test mass resting on its outer surface. Plotting the measure of effect on...
  40. C

    Could All Matter Be Curved Space Itself?

    Greetings, I read a lot. It is often repetitious. Occasionally I read an idea or way of looking at something that I had not read before. In one book, I read the matter did not curve space, but rather, matter *is* curved space. This is a paradigm shift in the way of thinking about matter...
  41. D

    Confusion with curved space analogies

    I've always been confused by the typical analogies I see when gravity as a space-time curvature is explained. In 2-D it is usually a plane with field lines, and the surface of the plane is curved around an object. And so we are told a mass placed in this curvature will "fall" down the curve...
  42. S

    Quantum Fluctuations and Curved Space

    There is a new theory being put forth that gravity may amplify vacuum energy to the point that the amplified vacuum energy may predominate over classical vacuum energy, which would cause it to influence astrophysical processes: http://www.physorg.com/news193330592.html It's just a...
  43. D

    Extrinsic properties of the curved space

    Example: take curved 2D space with positive constant curvature everywhere. You say, sphere with radius R? no, there are 2 different solutions in topology: sphere and half-sphere. Half sphere (1/2 of sphere where points across the 'equator' are connected to the opposite sides) can’t be 'embedded'...
  44. N

    Solving the Curved Space Puzzle: e or c?

    If you think of a bow (as in bow and arrow) placed in space in the proximity of a massive object. This bow for arguments sake is 100 km from tip to tip. Now replace the bow (wooden part and string) with what might be considered waypoints. The curved wooden part represents a straight line in...
  45. U

    Divergence theorem in curved space

    I have been contemplating my confusion about my intuition regarding GR and believe I have tracked down the primary source of confusion. The classical theories I have been taught assumed flat space with independent time and used the divergence theorem to derive inverse squared laws for fields...
  46. N

    Relativity, Gravity, Attraction , Curved Space

    Relativity, Gravity, "Attraction", Curved Space Okay, I am by no means knowledgeable in this field, probably more dangerous than anything. But I have a question about gravity. If two masses are identical (in an idea situation) and have no other forces acting on them, and start from a...
  47. P

    How does curved space create gravity?

    We've all seen the "ball on a rubber sheet" analogy, showing how warped space near a planet can cause a light beam to alter its path. We are told that the light is actually following the shortest path in curved space. When it comes to a *stationary* object near a planet, however, I have a...
  48. M

    The photon gas in the curved space

    In a flat space, the momentum of a photon gas distributes isotropically. Every direction is equivalent. If the space is curved,like the space outside a black hole, what will happen to the photon gas? Will the momentum distribution be not isotropic any more?
  49. N

    Acceleration and Curvature: Understanding the Relationship

    From another thread: (Bob for short's reply) I thought acceleration DID curve space... I'm coming from this perspective: say in the rotating "rigid" disc...and via Einstein's equivalence principle... for example, Brian Greene in THE ELEGANT UNIVERSE says: any clarifications...
  50. M

    Distinguishing Curved Space vs. Coordinate Choices in Mathematics

    How do we distinguish (mathematically) between curved space and the choice of coordinates? For example, the flat space metric in spherical polar coordinates looks as if it is curved space. I can ask the same for gravitational waves - how do we know that it isn't the TT gauge which is wavelike...