What is Spacetime metric: Definition and 29 Discussions

In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational potential of Newtonian gravitation. The metric captures all the geometric and causal structure of spacetime, being used to define notions such as time, distance, volume, curvature, angle, and separation of the future and the past.

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  1. 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...
  2. cianfa72

    I Schwarzschild spacetime in Kruskal coordinates

    As explained here in Kruskal coordinates the line element for Schwarzschild spacetime is: $$ds^2 = \frac{32 M^3}{r} \left( – dT^2 + dX^2 \right) + r^2 \left( d\theta^2 + \sin^2 \theta d\phi^2 \right)$$ My simple question is: why in the above line element are involved 5 coordinates and not just...
  3. cianfa72

    I Einstein Definition of Simultaneity for Langevin Observers

    Hi, reading this old thread Second postulate of SR quiz question I'd like to ask for a clarification on the following: Here the Einstein definition of simultaneity to a given event on the Langevin observer's worldline locally means take the events on the 3D spacelike orthogonal complement to...
  4. cianfa72

    I Minkowski Spacetime KVF Symmetries

    Hi, reading Carrol chapter 5 (More Geometry), he claims that a maximal symmetric space such as Minkowski spacetime has got ##4(4+1)/2 = 10## indipendent Killing Vector Fields (KVFs). Indeed we can just count the isometries of such spacetime in terms of translations (4) and rotations (6). By...
  5. 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...
  6. 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...
  7. 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...
  8. A

    A Dark matter and spacetime metric

    I'm wondering if the galactic rotation curves could be attributed to a deviation of the metric of spacetime from the ideal Schwarzschild metric. The Schwarzschild-metric is a well tested good approximation for the regions near the central mass - but at the outer space, far away from the...
  9. 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...
  10. LarryS

    I Spacetime Metric: Which signature is better?

    It seems that, in general, mathematicians prefer the (-,+,+,+) signature for the Minkowski spacetime metric while most physicists prefer the (+,-,-,-) signature. Is there any evidence that Nature actually prefers one over the other? As usual, thanks in advance.
  11. RyanH42

    What is the volume of the universe using a spacetime metric approach?

    I want to calculate two things (This is not a homework question so I am posting here or actually I don't have homework like this) First question is finding universe volume using spacetime metric approach.The second thing is find a smallest volume of a spacetime metric (related to plank...
  12. U

    Energy-Momentum Tensor of Perfect Fluid

    Homework Statement I am given this metric: ##ds^2 = - c^2dt^2 + a(t)^2 \left( dx^2 + dy^2 + dz^2 \right)##. The non-vanishing christoffel symbols are ##\Gamma^t_{xx} = \Gamma^t_{yy} = \Gamma^t_{zz} = \frac{a a'}{c^2}## and ##\Gamma^x_{xt} = \Gamma^x_{tx} = \Gamma^y_{yt} = \Gamma^y_{ty} =...
  13. U

    Frequency of Photon in Schwarzschild Metric?

    Homework Statement The schwarzschild metric is given by ##ds^2 = -Ac^2 dt^2 + \frac{1}{A} dr^2 + r^2\left( d\theta^2 + sin^2\theta d\phi^2 \right)##. A particle is orbiting in circular motion at radius ##r##. (a) Find the frequency of photon at infinity ##\omega_{\infty}## in terms of when it...
  14. U

    Satellite orbiting around Earth - Spacetime Metric

    Homework Statement The metric near Earth is ##ds^2 = -c^2 \left(1-\frac{2GM}{rc^2} \right)dt^2 + \left(1+\frac{2GM}{rc^2} \right)\left( dx^2+dy^2+dz^2\right)##. (a) Find all non-zero christoffel symbols for this metric. (b) Find satellite's period. (c) Why does ##R^i_{0j0} \simeq \partial_j...
  15. U

    Light-like Geodesic - What are the limits of integration?

    Homework Statement Consider the following geodesic of a massless particle where ##\alpha## is a constant: \dot r = \frac{\alpha}{a(t)^2} c^2 \dot t^2 = \frac{\alpha^2}{a^2(t)} Homework EquationsThe Attempt at a Solution Part (a) c \frac{dt}{d\lambda} = \frac{\alpha}{a} a dt =...
  16. 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...
  17. U

    How do I differentiate this Scalar Field?

    Homework Statement (a) Find the christoffel symbols (Done). (b) Show that ##\phi## is a solution and find the relation between A and B.[/B] Homework EquationsThe Attempt at a Solution Part(b) \nabla_\mu \nabla^\mu \phi = 0 I suppose for a scalar field, this is simply the normal derivative...
  18. U

    Lifetime of the universe - FRW

    Homework Statement [/B] (a) Find the value of A and ##\Omega(\eta)## and plot them. (b) Find ##a_{max}##, lifetime of universe and deceleration parameter ##q_0##. Homework Equations Unsolved problems: Finding lifetime of universe. The Attempt at a Solution Part(a)[/B] FRW equation is...
  19. U

    General Relativity - FRW Metric

    Homework Statement (a) Find the FRW metric, equations and density parameter. Express the density parameter in terms of a and H. (b) Express density parameter as a function of a where density dominates and find values of w. (c) If curvature is negligible, what values must w be to prevent a...
  20. U

    Einstein Equations of this metric

    Homework Statement [/B] (a) Find the christoffel symbols (b) Find the einstein equations (c) Find A and B (d) Comment on this metric Homework Equations \Gamma_{\alpha\beta}^\mu \frac{1}{2} g^{\mu v} \left( \partial_\alpha g_{\beta v} + \partial_\beta g_{\alpha v} - \partial_\mu g_{\alpha...
  21. U

    Solving this space-time Metric

    Homework Statement (a) Find ##\dot \phi##. (b) Find the geodesic equation in ##r##. (c) Find functions g,f,h. (d) Comment on the significance of the results. Homework Equations The metric components are: ##g_{00} = -c^2## ##g_{11} = \frac{r^2 + \alpha^2 cos^2 \theta}{r^2 +\alpha^2}##...
  22. U

    Quick question on Geodesic Equation

    Starting with the geodesic equation with non-relativistic approximation: \frac{d^2 x^{\mu}}{d \tau^2} + \Gamma_{00}^{\mu} \left( \frac{dx^0}{d\tau} \right)^2 = 0 I know that ## \Gamma_{\alpha \beta}^{\mu} = \frac{\partial x^{\mu}}{\partial y^{\lambda}} \frac{\partial^2 y^{\lambda}}{\partial...
  23. U

    General Relativity - Circular Orbit around Earth

    Homework Statement (a) Find the proper time in the rest frame of particle (b) Find the proper time in the laboratory frame (c) Find the proper time in a photon that travels from A to B in time P Homework EquationsThe Attempt at a Solution Part(a) [/B] The metric is given by: ds^2 =...
  24. R

    How does one get time dilation, length contraction, and E=mc^2 from spacetime metric?

    How does one get time dilation, length contraction, and E=mc^2 from the spacetime metric? Suppose all that you are given is x12 + x22 + x32 - c2t2 = s2 How do you derive time dilation, length contraction, and E=mc^2 from this? What is the most direct way to do this?
  25. M

    Anti-de Sitter spacetime metric and its geodesics

    Hello, everybody. I have some doubts I hope you can answer: I have read that the n+1-dimensional Anti-de Sitter (from now on AdS_{n+1}) line element is given, in some coordinates, by: ds^{2}=\frac{r^{2}}{L^{2}}[-dt^{2}+\sum\limits_{i=1}^{n-1}(dx^{i})^{2}]+\frac{L^{2}}{r^{2}}dr^{2} This can be...
  26. L

    Exploring Spacetime Metric for Clock A's Time Interval

    Consider the spacetime metric ds^2=-(1+r)dt^2+\frac{dr^2}{(1+r)} + r^2 ( d \theta^2 + \sin^2{\theta} d \phi^2) where \theta, \phi are polar coordinates on the sphere and r \geq 0. Consider an observer whose worldline is r=0. He has two identical clocks, A and B. He keeps clock A with...
  27. W

    Does Time Variation Necessarily Imply Full Spacetime Metric?

    Background: Math: An affine parameter provides a metric along a geodesic but not a metric of the space, for example between geodesics. A connection provides an affine parameter, and a non-trivial connection gives rise to Riemann curvature. Given the existence of a connection with Riemann...
  28. D

    Curl added to the spacetime metric.

    Hello! I was thinking the other day, of the Earth's rotation around its axis. If one spins a boiled egg, it maintains spin longer than does an unboiled. Eventually both stops because of the friction against the floor, but not at the same time. The Earth has different levels of viscosity...
  29. D

    What is the Spacetime Metric and its Describing Equations in Layman's Terms?

    I'm a layman here, so please put any answers in terms that a layman can understand. You can use calculus though :) What is the spacetime metric, and what are the equations describing it?