What is Curvature of spacetime: Definition and 44 Discussions

General relativity, also known as the general theory of relativity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time or four-dimensional spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of partial differential equations.
Some predictions of general relativity differ significantly from those of classical physics, especially concerning the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light. Examples of such differences include gravitational time dilation, gravitational lensing, the gravitational redshift of light, the gravitational time delay and singularities/black holes. The predictions of general relativity in relation to classical physics have been confirmed in all observations and experiments to date. Although general relativity is not the only relativistic theory of gravity, it is the simplest theory that is consistent with experimental data. Unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity; and how gravity can be unified with the three non-gravitational forces—strong, weak, and electromagnetic forces.
Einstein's theory has important astrophysical implications. For example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. There is ample evidence that the intense radiation emitted by certain kinds of astronomical objects is due to black holes. For example, microquasars and active galactic nuclei result from the presence of stellar black holes and supermassive black holes, respectively. The bending of light by gravity can lead to the phenomenon of gravitational lensing, in which multiple images of the same distant astronomical object are visible in the sky. General relativity also predicts the existence of gravitational waves, which have since been observed directly by the physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of a consistently expanding universe.
Widely acknowledged as a theory of extraordinary beauty, general relativity has often been described as the most beautiful of all existing physical theories.

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  1. Sciencemaster

    I Calculate Gaussian Curvature from 4D Metric Tensor

    I've been trying to find a way to calculate Gaussian curvature from a 4D metric tensor. I found a program that does this in Mathematica using the Brioschi formula. However, this only seems to work for a 2D metric or formula (I would need to use something with more dimensions). I've found...
  2. 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?
  3. cianfa72

    I Global simultaneity surfaces - how to adjust proper time?

    Hi, searching on PF I found this old post Global simultaneity surfaces. I read the book "General Relativity for Mathematicians"- Sachs and Wu section 2.3 - Reference frames (see the page attached). They define a congruence of worldlines as 'proper time synchronizable' iff there exist a...
  4. cianfa72

    B Basic introduction to gravitation as curved spacetime

    Hi, my daughter saw my MTW copy on the desk and she asked me about the picture with the apple in front. To introduce her to the idea of gravitation as curved spacetime I answered like this: Consider you (A) and a your friend (B) at two different spots on a garden each with a firecracker. Take...
  5. 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?
  6. Martian2020

    B Visualize 2D Intrinsic Curvature of Spacetime (1s+1t) in 3D

    Via web search found https://www.physicsforums.com/threads/what-dimension-does-space-time-curve-in.852103/ Read it and watched two videos mentioned: I understand we cannot perceive 5D ;-), so extrinsic visualization of maximum of 2D intrinsic curvature is possible. So time+1d space is all we...
  7. E

    Space With Schwarzschild Metric

    This is a problem from Tensor Calculus:Barry Spain on # 69 Prove that a space with Schwarzschild's metric is an Einstein space, but not a space of constant curvature. The metric as given in the book is $$d\sigma^2=-\bigg(1-\frac{2m}{c^2r}\bigg)^{-1}dr^2-r^2d\theta^2-r^2\sin^2 \theta...
  8. 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...
  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. DAirey

    I Does a metric exist for this surface?

    I have a surface defined by the quadratic relation:$$0=\phi^2t^4-x^2-y^2-z^2$$Where ##\phi## is a constant with units of ##km## ##s^{-2}##, ##t## is units of ##s## (time) and x, y and z are units of ##km## (space). The surface looks like this: Since the formula depends on the absolute value of...
  11. 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...
  12. 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...
  13. SD das

    I Consequence of curvature of spacetime

    I was wondering that consequence of curvature of spacetime caused by an entire galaxy! Can anyone suggest?
  14. S

    I Solving Confusion About Parallelograms in Curved Spacetime

    One way to see that spacetime is curved is to try and draw a "rectangle" in spacetime (see the figure in the Feynman lectures, ch 42.7): If I wait for 100 seconds and then move upwards on earth, I end up at a different point in spacetime than when I first move upwards and then wait for 100...
  15. S

    I Number of independent components of the Riemann tensor

    I've thought of a new way (at least I never read it anywhere) of counting the independent components of the Riemann tensor, but I am not sure whether my arguments are valid, so I would like to ask whether my argument is sound or total bonkers. The Riemann tensor gives the deviation of a vector A...
  16. arupel

    B Is Space Infinite? Examining the Big Bang Theory and the Curvature of Spacetime

    From what I read attempts to measure the curvature of space have not succeeded. It would seem there may not be a curvature of space time. If this is true then what may be implied is that space goes on forever. If this is true how could the big bang theory, if it could, give a reasonable answer...
  17. Tony Stark

    Curvature of spacetime inside hollow sphere

    If mass curves spacetime in its vicinity, then consider the following case- Take a heavy hollow lead sphere which has 2 smaller lead balls placed in it. The Outer Sphere will curve spacetime around itself and thus will have its own gravity, but what about the 2 balls placed in it? The spacetime...
  18. S

    Graviton vs Einstein's Curvature of Spacetime

    As far as I understand it, Einstein theorized that gravity was the result of the curvature of space created by the presence of mater/energy, but that idea seems like it does not meld well with the idea that gravity is the result of a specific force carrying particle, as with the other...
  19. RisingSun361

    How does the curvature of spacetime affect space?

    If the fabric of the universe is made of both space and time, and curving spacetime affects time, then I'm guessing it also affects space. I'm aware that an object shortens in length as it approaches the speed of light. But in the case of gravity, is space relative like time? Does an object on...
  20. R

    Ball Thrown Up Follows Curvature of Spacetime

    A ball thrown up comes down again because the ball follows the curvature of spacetime. So is it possible that the ball comes down and before hitting the surface, goes up again due to curvature of the spacetime,which now point upward.
  21. R

    If gravity is curvature of spacetime, why unified force @ Planck epoch

    Hi Bear with my possible ignorant. I am puzzled over this dilemma. If General Relativity states that gravity is the curvature of spacetime, that is, no spacetime no gravity, and the cause of curvature is matter (mass), it means that if no matter, there is no gravity. I understand that...
  22. B

    Why the curvature of spacetime is related to momentum?

    Well, I'm totally in a mess now
  23. J

    Does the curvature of spacetime have a unit?

    Does the curvature of spacetime have a unit?
  24. S

    Why do objects fall through curvature of spacetime?

    Just wondering, if the way to describe the movement of objects through spacetime is to say that they fall through the curves created in 4D spacetime, then is it a stupid question to ask why objects don't rise through spacetime? Or is this the same thing and rising and falling are one of the same...
  25. K

    Understanding Positive and Negative Intrinsic Curvature in General Relativity

    if gravity arises from normal accelerations due to the curvature of spacetime...what would the opposite of this "process" represent? to clarify is it possible to describe the opposite of this curvature?? thanks
  26. M

    Curvature of Spacetime: Is the Universe Flat or Curved?

    Hi there. I have a dump question for you guys. I really wonder about curvature of spacetime. I read that due to Omega_tot=1 the Universe is assumed to be flat. But on the other hand something like the curvature of the universe is mentioned... I also thought that the energy stress tensor...
  27. V

    Question about the curvature of spacetime:

    There's something very fundamental about the curved structure of spacetime that is confusing me. Einstein is saying that gravity can bend starlight. In other words, if I have this right, a star's light will follow the curvatures of spacetime created by a large body of mass, like the sun. Here's...
  28. S

    Question: does the physical curvature of spacetime ever move ?

    Question: does the physical curvature of spacetime ever "move"? Something isn't adding up with Einstein's theory--or, more likely, I'm just not understanding it correctly! How can we say that the curvatures of spacetime created by the presence of stress-energy is giving us a continuum? When I...
  29. J

    Curvature of Spacetime on Earth

    I am trying to improve my understand of the basic elements of GR. I have read that the Earth orbits the sun because spacetime between the Earth and the sun is warped, mainly due to the sun’s mass. The Earth follows a geodesic, which is the equivalent of a straight line in curved space...
  30. P

    Curvature of Spacetime related to mass and expansion

    Hi, I was wondering if anyone could clarify something for me. I have been reading about the curvature of Spacetime and have come across a few things in articles in conjunction with de Sitter and Anti de Sitter spaces "Negative curvature corresponds to an attractive force" and "Positive...
  31. R

    How does mass cause the curvature of spacetime at a distance?

    Since it's been observed that mass causes the curvature of the spacetime continuum. I'm wondering how it curves the spacetime continuum at a distance. For example, a planet will curve the spacetime around it millions of miles away, yet all around the planet is the almost perfect vacuum of...
  32. D

    The unit vector normal to the curvature of spacetime

    Let us say there is a curved region of spacetime whose curvature is \kappa(s). How does one find the coordinates of the unit vector normal to a certain point on the region of spacetime? I tried searching Hamilton's principle and the general theory of relativity but I could not find any equation...
  33. Islam Hassan

    Gravitons vs Attraction Due to Curvature of Spacetime

    If one day gravitons are discovered, would their action be complementary to the gravitational attraction due to curved spacetime? Can gravity arise from both curved spacetime and exchange of gravitons? IH
  34. Y

    Device to measure curvature of spacetime.

    While posting a reply in another thread, I had an inspiration for a device to measure spacetime curvature. It is well know that we can measure this curvature by measuring the angles of a large triangle or comparing the circumference of a circle to its radius, but his device may may be simpler or...
  35. R

    Curvature of spacetime inside of stars and planets

    hi, how does general relativity work INSIDE stars and planets, since the mass is no longer concentrated within a point, so there are necessarily gravitationnal effects outwards and not only inwards?
  36. T

    Does the curvature of SpaceTime Disprove gravity?

    Obviously not. But I am confused on this. Newtons laws of gravity did not fit with Einsteins Relativity theories because Newton said that a change in matter of an object would cause a shift to be felt by others in the universe because of the change in gravity. This would mean that the movement/...
  37. G

    Curvature of spacetime, Matter vs Energy

    Hello, I have what is probably a relatively (no pun intended) simple question pertaining to general relativity. I thought that I had achieved a solid understanding of the theory (and its special counterpart), until this question formed in my mind. My problem is based on the following...
  38. F

    What's the curvature of spacetime describe?

    Does the curvature of spacetime describe gravity or tidal force?
  39. R

    Curvature of Spacetime: an inacurate Description?

    I think about general relativity often, specifically about the curvature of spacetime in the presence of matter (gravity). For a while, I understood much of this concept, but certain things escaped me: when objects are moving, it is easy to see how curved space causes matter to move in the way...
  40. Y

    What does curvature of spacetime really mean?

    I don't really get GR. Why should curved space and time be a model for gravity? To me, curved space means a observers no longer measure distances as sqrt(x^2+y^2+z^2), but rather, given an x-ordinate, y-ordinate and z-ordinate, the length of the shortest path to that coordinate can be calculated...
  41. J

    Black Holes, Quantum Gravity and the Curvature of spacetime

    What is Quantum Gravity and the Curvature of Spacetime and how is it all relevant to one another?
  42. H

    Dark Energy & Curvature of Spacetime

    General relativity says that the gravitational "field" is just the warping of space by mass. I like to think of the ball on the trampoline analogy. Is dark energy, basically negative pressure, be caused by the natural curvature of spacetime? http://www.geocities.com/ixi_dima_ixi/gr.JPG
  43. Mk

    Question about cause of curvature of spacetime

    Is gravity a result of the curvature of spacetime, or is it the curvature of spacetime a result of gravity? Or something else? Thank you.
  44. R

    What Causes the Curvature of Spacetime for Orbiting Bodies?

    we are doing forces and gravity , but what I don't understand is when the sun curves spacetime for the Earth to orbit this curvature, what causes it ? thanks in advance. roger
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