Spacetime curvature Definition and 88 Threads

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. cianfa72

    I SR-time dilation vs GR-time dilation on rotating Earth

    Reading this old thread does-special-and-general-relativity-affect-aging-of-apollo-astronauts I came with some question. To me isn't clear how SR-time dilation by rotation of the Earth is compensated by GR-time dilation effect.
  2. cianfa72

    I Congruence of curves in spacetime

    Hi, I asked a similar question in Differential Geometry subforum. The question is related to the symmetries of spacetime. Namely take a spacetime with a timelike Killing vector field (KVF). By definition it is stationary. Now consider a curve ##\alpha## and translate it along the KVF by a...
  3. B

    I How Does Mass Influence Spacetime Curvature?

    I have read widely about spacetime but not deeply. I am not good with high mathematics. I am aware of and comfortable with the concept: “matter tells spacetime how to curve, and curved spacetime tells matter how to move.” But not much deeper. This is a thought experiment. Is there an equation...
  4. cianfa72

    I GPS system and general relativity

    Hi, we had a thread some time ago about GPS satellite system. One starts considering the ECI coordinate system in which the Earth's center is at rest with axes pointing towards fixed stars. One may assume it is an inertial frame in which the Earth's surface undergoes circular motion. Clocks on...
  5. cianfa72

    I Spacetime coordinate smoothness requirement

    Hi, I was keep reading the interesting book Exploring Black Holes - second edition from Taylor, Wheeler, Bertschinger. I'd like to better understand some points they made. In Box 3 section 3-6 an example of coordinate singularity at point O in Euclidean plane in polar coordinates centered there...
  6. cianfa72

    I Gravitational wave propagation in GR

    Hi, I'd like to discuss in this thread the propagation of Gravitational Waves (GW) in the context of GR. Just to fix ideas, let's consider a FW spacetime. It is not stationary (even less static), however the timelike congruence of "comoving observers" is hypersurface orthogonal. Suppose at a...
  7. cianfa72

    I Meaning of the distance between galactic objects in GR

    Hi, I was thinking about the claim that for instance Sagittarius A* (Sgr A*) black hole is a at 26996±29 light years from the Earth from a GR point of view. Assuming a FLRW model for the Universe, maybe the above meaning is that at a given cosmological time ##t## (the "present" time) the proper...
  8. cianfa72

    I Macroscopic objects in free-fall

    Hi, very basic question. Take an object like a rock or the Earth itself. If we consider their internal constituents, there will be electromagnetic forces acting between them (Newton's 3th law pairs). From a global perspective if the rock is free from external non-gravitational forces, then it...
  9. jaketodd

    I Effects of magnetism on spacetime curvature, and implications

    Quotes from Cosmic magnetism, curvature and the expansion dynamics: "Most interestingly, the coupling between magnetism and geometry implies that even weak fields have a significant impact if the curvature contribution is strong." "The energy scales involved vary from ∼ 100 MeV at the QCD...
  10. luro1964

    I Did Born Rigidity Lead Einstein to Conclude Spacetime is Curved?

    I'll set out by saying that I have no real formal training in physics or maths. However, I have been keen to try to understand what exactly convinced Einstein that spacetime must be curved. As I understand it, the bending of star light was already explained by Newtonian physics, although of...
  11. Marcarious Thomas

    B Faster Than Light Travel: Exploring the Possibilities of Spacetime Curvature

    Based on the current understanding of general relativity, it is possible that curving spacetime in the back of a spacecraft would allow for faster-than-light travel. In general relativity, the curvature of spacetime is determined by the universe's distribution of matter and energy. If a...
  12. Structure seeker

    I Research on conservation of spacetime curvature

    After trying to kinda get a picture of the field of play in quantum physics according to the standard model, a question came up. I tried to formulate the known bosons each as a particle transferring some property. 1. Photons transfer electric charge: the electromagnetic force gives attraction...
  13. Narasoma

    I Spacetime Curvature via Triangle

    I understand the mechanism of defining the curvature of a 2D manifold via triangle. But I don't understand how this works in 3D. Meanwhile, Lawrence Krauss mentioned in his book A Universe from Nothing it does. How does this work in 3D?
  14. cianfa72

    I Coord. Time Vector Field: Schwarzschild vs Gullstrand-Painleve

    Hi, I was reading this insight schwarzschild-geometry-part-1 about the transformation employed to rescale the Schwarzschild coordinate time ##t## to reflect the proper time ##T## of radially infalling objects (Gullstrand-Painleve coordinate time ##T##). As far as I understand it, the vector...
  15. cianfa72

    I Synchronous Reference Frame: Definition and Usage

    Hi, reading the Landau book 'The Classical theory of Field - vol 2' a doubt arised to me about the definition of synchronous reference system (a.k.a. synchronous coordinate chart). Consider a generic spacetime endowed with a metric ##g_{ab}## and take the (unique) covariant derivative operator...
  16. cianfa72

    I Is acceleration absolute or relative - follow up

    Hello, Some doubt arose me reading this thread https://www.physicsforums.com/threads/is-acceleration-absolute-or-relative-revisited.999420/post-6454462 currently closed. Sorry, I have not be able to quote directly from it :frown: Your claim is not , however, asserting that the spacetime...
  17. cianfa72

    I The spacetime length of finite spacelike intervals

    Hello, I'm aware of the following topic has already been discussed here on PF, nevertheless I would like to go deep into the concept of "finite spacelike interval" in the context of SR and GR. All us know the physical meaning of timelike paths: basically they are paths followed through...
  18. K

    B Explaining Spaghettification w/ Spacetime Curvature

    I enjoy explaining spacetime curvature to people with a rank-beginner understanding of GR. But someone asked about that favorite concept in pop-sci, spaghettification. I'm having a hard time with it. If you fell into a black hole, there's no reference frame within which you could describe...
  19. cianfa72

    I Mercury Perihelion Precession: Analytic Derivation

    HI, I'm curios about the analytic derivation of Mercury perihelion precession starting from EFE - Einstein Field Equation (or simply just from Schwarzschild solution of the EFE). Can you advise me about some source or online material to learn it ? Thanks.
  20. cianfa72

    I Spacetime distance between spacelike related events

    Hi, in general relativity I'm aware of the spacetime 'distance' between two timelike related events is maximized by the free falling timelike path (zero proper acceleration) joining them. Consider now a couple of events belonging to a spacelike hypersurface (AFAIK it is an hypersurface with...
  21. cianfa72

    I Gravitational time dilatation and curved spacetime - follow up

    Hi, starting from this very interesting thread I'm still a bit confused about the conclusions. The main point, as far as I can understand, is all about conditions for a quadrilateral to be considered a parallelogram. My first basic doubt is: the concept of 'parallel' applies just to geodesic...
  22. Ranku

    I Spacetime curvature and curvature index

    The presence of the cosmological constant produces a flat spacetime universe with Ω = 1. There is also the curvature index of space k, which can be +1, 0, -1. But it is possible to have any of these values of k with Λ > 0 or Λ < 0. How is the curvature of spacetime determined by Λ different from...
  23. Osvaldo

    I Spacetime Curvature: Eliptical Orbital Paths & Keppler Laws

    Though it is hard not to believe in the spacetime curvature that cause planets to follow curved path arround massive objects, I wander how come these paths are eliptical, the object change velocity when moving arround the massive object and what is more obeys the Keppler laws. If there is not...
  24. 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...
  25. cianfa72

    I About spacetime coordinate systems

    Hi, There is a point that, in my opinion, is not quite emphasized in the context of general relativity. It is the notion of spacetime coordinate systems that from the very foundation of general relativity are assumed to be all on the same footing. Nevertheless I believe each of them has to be...
  26. 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...
  27. 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...
  28. O

    B Can We Stop Gravity? Exploring the Possibility

    I know this is probably going to sound stupid but I'm really curious for the opinion of someone who has knowledge. Well spacetime is like a stretched bed linen. Putting on an object with a big mass, will cause its curvature. Now imagine having a piece of fabric holding it stretced from both...
  29. S

    I Torsion & Non-Closed Rectangle in Feynman & Penrose

    In the Feynman Lectures on Physics, Feynman explains the curvature of spacetime by drawing a rectangle in spacetime, see http://www.feynmanlectures.caltech.edu/II_42.html Fig. 42.18 First waiting 100 sec and then moving 100 feet in height on Earth's surface results in a different situation...
  30. L

    I GR and its bending of space time

    Hello Maybe my question is dumb but is the bent of space time instant due to gravity? If a mass pops into existence will space time be bent instantly ? Intercations between forces are light speed but gravity is and is not a force depending on pov
  31. 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...
  32. 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...
  33. 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...
  34. PeterDonis

    A Does gravitational time dilation imply spacetime curvature?

    In a recent thread, the question came up of whether the presence of gravitational time dilation implies spacetime curvature. My answer in that thread was no: This was based on the obvious counterexample of observers at rest in Rindler coordinates in flat Minkowski spacetime; two observers at...
  35. Aikiman007

    B Warping Spacetime: Classical Physics & Relativity

    Based on classical physics all things attract one another due to their own gravity pull, so theoretically the Earth is drawn closer to a tennis ball even if its impossible to detect. Knowing this and Einsteins theory of relativity of spacetime could I not assume then that every physical body...
  36. arupel

    A How was the value of the spacetime curvature obtained?

    I received a reply about the value of space time curvature. Evidently it is not zero but an almost unconceiveably small number. I was just wandering how this number was gotten.
  37. P

    B How do we study space-time singularities?

    I hope I am posting this in the correct forum, I am trying to better understand space-time singularities. I can find easily the basic, and advanced information on what it is and the different theories. My main question is how do scientists study these space-time singularities? Thank you for...
  38. V

    I Measuring Coordinates in Strong Gravity: Schwarzschild Metric

    We know that Schwarzschild metric describes an asymptotically flat spacetime. This means that far away from the event horizon we can safely interpret the ##r## coordinate as distance from the center. But when close enough to the event horizon the curvature becomes significant and our common...
  39. A

    I Quasars: What Metric Should You Use?

    Hello I just got interested into quasars and I have a question:What metric do you use for quasars?
  40. A

    I Solving Geodesic Equations with Killing Vectors: Is There a General Solution?

    Hello I am concered about way of solving geodesic equation. Is there a general solution to geodesic equation? How to calculate the Cristoffel symbol at the right side of the equation? Thanks for helping me out!
  41. A

    I What is the velocity of the photon through the fourth dimension x4?

    What is the velocity of the photon through the fourth dimension x4? Photons are real, physical entities. The fourth dimension is a real, physical entity. Therefore, photons must have a relationship with the fourth dimension. They must have some velocity relative to it. What is the velocity...
  42. F

    I Black Hole & Spacetime Curvature: Can Mass Decrease It?

    Since a black hole goes to a singularity, theoretically wouldn't added mass to that point decrease the spacetime curvature by increasing of the circumference, and then not have a loss in information.
  43. Augbrah

    Projecting push forward of a vector

    Homework Statement Say we have two manifolds N(dim d) and M(dim d-1). Let Φ: M →N be a diffeomorphism where Σ = Φ[M] is hypersurface in N. Let n be unit normal field (say timelike) on Σ and ⊥ projector (in N) defined by: ⊥^a_b = \delta^a_b + n^a n_b Where acting on (s, 0) tensor projection...
  44. R

    How could the infant universe have existed as a singularity?

    A singularity is a region in which the curvature of space-time becomes infinite. But according to standard big bang models, at the initial point (at which T = 0) the pre-expansion space - as miniscule as it was - was filled uniformly with all energy that ever existed or will exist. But if all...
  45. K

    I Exploring Einstein's Theory of Gravity: Time Dilation and Tidal Forces

    I have some questions related to this video: In the Einstein view of gravity, time is warped. Is this warped time same as the gravitational time dilation? In other words, is the curved time axis due to different clock speeds at different height in a gravitational field? Further, can the tidal...
  46. K

    Space-Time Curvature in General Relativity

    Suppose we are in a Minkowskian space, away from all the source of gravity, and observe an accelerated frame from this frame. Acoording to Equivalence principle, we can consider the accelerated frame to be at rest and assume we have gravity in the accelerated frame. Thus, observer in the...
  47. K

    Confusion in General Relativity

    According to equivalence principle, gravity can be treated like acceleration "locally". Based on this principle we can treat a non-inertial frame at rest and explain the fictitious forces (of Newton's Laws) as gravity. From this we can prove that time elapses at different rates at different...
  48. G

    Covariant fields and new physics

    I assume that all the fundamental physics known - as of today - can be reduced to quantum general covariant fields (including spacetime itself to be seen as a field of those...). Now, sorry if my question is quite abstract and based on tomorrow's hypothetical new physics, but would it be against...
  49. T

    Heory of Spacetime Curvature and Gravity

    Do gravitational forces have to follow spacetime in the same way as light? Or does gravity act in a higher dimension? Thanks, T
  50. S

    Gravity & Spacetime Curvature: Have I Understood?

    When spacetime is not bent the two objects, red ball and blue ball, will move strait up the y-axis as they move through time. (Space is x and time is y). Now I've made the assumption that either a) All things want to move the smallest possible distance to the next point in time or b) all...
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