What is General relativity: Definition and 999 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.
I'm utterly confused by co-moving distance, transverse comoving distance and proper distance. Is comoving distance = proper distance? Then what is transverse comoving distance? Here's what I know so far:
The FRW metric can either be expressed as
ds^2 = c^2dt^2 - a^2(t) \left[ \frac{dr^2}{1-kr^2}...
Hi, I have been studying general relativity using Hobson's lately, particularly about the FRW universe.
I know that for a matter universe with curvature,
H^2 = \left( \frac{\dot a}{a} \right)^2 = \frac{8\pi G}{3} \rho_m -\frac{kc^2}{a^2}
Another expression I came across is also
1 = \Omega_m +...
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...
I was reading through hobson and my notes where the covariant acts on contravariant and covariant tensors as
\nabla_\alpha V^\mu = \partial_\alpha V^\mu + \Gamma^\mu_{\alpha \gamma} V^\gamma
\nabla_\alpha V_\mu = \partial_\alpha V_\mu - \Gamma^\gamma_{\alpha \mu} V_\gamma
Why is there a minus...
Homework Statement
(a)Find how ##\rho## varies with ##a##.
(b) Show that ##p = \frac{2}{\lambda^2}##. Find ##B## and ##t_0##.
(c) Find ##w## and ##q_0##. What values of ##\lambda## makes the particle horizon infinite? Find the event horizon and age of universe.
(d) Find luminosity distance...
Homework Statement
A 26 year-old biologist makes a trip to study alien life forms in a distant planet 10 light-years away. The round trip including a stay of 1 year in the distance planet takes 21.5 years according to the clock on earth. The biologist’s son is 3 year-old when she left. Assume...
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...
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...
I'm reading An Intro to GR, Hughston and Tod, it says that in GR the idea is that the geometry of st varies from point to point, represented by allowing the metric to vary over space-time.
It uses a (+,-,-,-) signature and so ##proper time=ds^{2}##.
It then makes the comment that proper time...
HI - we know that the orbit of mercury precesses (I hope I am using the right terminology here). Which basically means that the orbit seems to undergo some sort of rotation in the ecliptic plane. Does this also mean that the period of Mercury's orbit as seen from the Earth is not uniform but...
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...
Is anyone familiar with Atlas 2 for mathematica to calculate the Riemann Tensor, Ricci Tensor, and scalar I have a metric that I need to calculate these things for. Can anyone help? I'm not too up on Mathematica either.
I have been looking online for books on introductory level quantum mechanics and General relativity that provide a mathematical introduction to these theories. Most of the books I have read until now provide a laymans introduction to these things.
Since I'm only pursuing this as a hobby and...
Taken from Hobson's book:
How is this done? Starting from:
R_{abcd} = -R_{bacd}
Apply ##g^{aa}## followed by ##g^{ab}##
g^{aa}g^{aa} R_{abcd} = -g^{ab}g^{aa}R_{bacd}
g^{ab}R^a_{bcd} = -g^{ab}g^{aa}R_{bacd}
R^{aa}_{cd} = - g^{ab}g^{aa} R_{bacd}
Applying ##g_{aa}## to both sides...
In a previous thread, reference was made to an entertaining "defense" of relativity by Einstein, which can be found here:
https://en.wikisource.org/wiki/Dialog_about_Objections_against_the_Theory_of_Relativity
One of the arguments Einstein makes in this dialog is that the twin paradox can be...
Assume we have a free-falling particle in gravity in a static metric. Its worldline is described by:
g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}
where ##|h_{\mu \nu} << 1|##.
Taken from Hobson's book:
Why did they let ##g^{k\mu} = \eta^{k\mu}##?
Hi guys!
I am reading the book "Gravity" by Hartle. I came across this scary-looking integral. The author does integration by parts and I don't get how he does it. Could someone guide me please?
Relevant equations:
∫u dv = uv - ∫v du
Guys , i have just completed my high school (4 days back) and know just the basic maths and physics (only what was taught in school).I have a great interest in physics and want to become a researcher.I am free for 3-4 months (after which i will join a college for majors in physics) and in that...
Are the general principles of string theory “the properties of fundamental particles are due to their unique vibrational pattern of strings in 10 or 11 or 26 dimensions” incompatible with both general and special relativity? due to an observer would be able to detect changes in properties of...
I stumbled upon this video:
which seems to claim that past present and future exist at the same time due to the fact that observes in different frames would not agree on the order of events. The video claims that there are frames in which Mozart is alive.
I don't think that this is correct...
Hello everyone,
I'm studying some applications of AdS/CFT and I came across an expression of the Maxwell field written in the following way:
$$
A=A_t(r)dt+B(r)xdy.
$$
How does this notation work? Is it simply a way of writing the four-vector? If so, why do we use this notation?
Thanks a lot!
I have seen written out in various places (including this forum) the effective potential function that comes from the solutions to the Schwarszschild Geodesic. But I haven't been able to find the effective potential functions for other solutions to Einstein's field equations. Are there...
http://en.wikipedia.org/wiki/Kaluza–Klein_theory
##http://link.springer.com/article/10.1007/BF01390677 (original german paper, Ich kan nicht Deutsche)
http://www.scientificexploration.org/journal/jse_21_3_beichler.pdf (this author makes some interesting arguments)
Also, a lot (if not all...
Homework Statement
(a) Show that the equations satisfy FRW equations.
(b) Show the metric when ##\eta## is taken as time
Homework EquationsThe Attempt at a Solution
[/B]
The FRW equation is:
3 \left( \frac{\dot a}{a} \right)^2 = 8\pi G \rho
Using ##\frac{da}{dt} = \frac{da}{d\eta}...
What do they mean by 'Contract ##\mu## with ##\alpha##'? I thought only top-bottom indices that are the same can contract? For example ##A_\mu g^{\mu v} = A^v##.
I have been reading section 3.1 of Wald's GR book in which he introduces the notion of a covariant derivative. As I understand, this is introduced as the (partial) derivative operators \partial_{a} are dependent on the coordinate system one chooses and thus not naturally associated with the...
Homework Statement
(a) Show acceleration is perpendicular to velocity
(b)Show the following relations
(c) Show the continuity equation
(d) Show if P = 0 geodesics obey:
Homework EquationsThe Attempt at a SolutionPart (a)
U_{\mu}A^{\mu} = U_{\mu}U^v \left[ \partial_v U^{\mu} +...
Homework Statement
[/B]
(a) Find the proper distance
(b) Find the proper area
(c) Find the proper volume
(d) Find the four-volume
Homework EquationsThe Attempt at a Solution
Part (a)
Letting ##d\theta = dt = d\phi = 0##:
\Delta s = \int_0^R \left( 1-Ar^2 \right) dr = R \left(1 -...
Homework Statement
Find the deflection of light given this metric, along null geodesics.
Homework EquationsThe Attempt at a Solution
[/B]
Conserved quantities are:
e \equiv -\zeta \cdot u = \left( 1 - \frac{2GM}{c^2r} \right) c \frac{dt}{d\lambda}
l \equiv \eta \cdot u = r^2 \left( 1 -...
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 =...
I was reading my lecturer's notes on GR where I came across the geodesic equation for four-velocity. There is a line which read:
Summing them up,
\partial_i g_{aj} u^i u^j - \frac{1}{2} \partial_a g_{ij} u^i u^j = \frac{1}{2} u^i u^j \partial_a g_{ij}
I'm trying to understand how LHS = RHS...
We know that when a rigid frame, say a rocket undergoes constant proper acceleration, its worldline is hyperbolic. The equation is given by:
x^2 - c^2t^2 = \left( \frac{c^2}{a_0} \right)^2
Suppose P is such a worldline and worldine can also be written as:
I understand how these are...
Homework Statement
(a) Find matrix element ##M_{ij}##
(b) Show that ##x^j## is an eigenvector of ##M_{ij}##
(c) Show any vector orthogonal to ##x^j## is also an eigenvector of ##M_{ij}##
Homework EquationsThe Attempt at a Solution
Part(a)
[/B]
\frac{\partial^2 \Phi}{\partial x^i x^j} =...
I have studied most of the concepts of Newtonian mechanics and understood the calculations and derivations involving the concerned formulae.
I have a basic knowledge of calculus (differentiation of products along with differentiation of trigonometric, logarithmic, exponential and implicit...
I am trying to derive the gravitational red shift effect but I think I am going about it all wrong. Specifically, I want to derive the change in frequency/ wavelength when a photon moves away from the surface of a star mass M and radius R.
So I tried to use relativistic mass of the photon and I...
I read in a magazine (namely Scientific American) that the Standard Model successfully combines Quantum Mechanics with General Relativity, but I also remember reading in The Elegant Universe that the Standard Model fails to do so. What's true and what's not?
Hi all, I'm fairly new to GR, and I'm also somewhat new to tensors as well. I'm looking for some detailed explanation of a tensor, as I want to begin studying GR mathematically. I watched a video that was posted on PF not too long ago that was pretty good. I'm having trouble remembering who it...
Hello, Everybody!
I'm new to the board, and am happy to have found you!
I have six questions I can't seem to find the answers to, either in books or online -- yet I know the answers are out there! I'll post each question in a separate thread, and hope that someone who knows far more than I do...
I understand that all physical laws essentially codify mathematically observed behavior. Newton codified Kepler and Brahe data, for example. Quantum Mechanics codifies observed particle behavior at relatively low speeds, etc. But Einstein had no empirical data to work from… So, I do not...
Homework Statement
Suppose everything is moving slowly, How can we find the metric tensor in GR in terms of the mass contained.
Homework Equations
I understand in case of everything moving slowly only below equation is relevant -
R00 - ½g00R = 8πGT00 = 8πGmc2
The Attempt at a Solution
None.
In considering special relativity as a limiting case of the general theory (without matter or curvature) the question arose as to whether the pseudo-riemann nature of the SR metric is actually an independant (essentially experimentally determined) assumption/property or derivable from the...
Homework Statement
Close to a Schwarzschild black hole, a photon is emitted between r = 2(mu) and 3(mu), where \mu = \frac{GM}{c^2} . The photon is emitted at an angle (alpha) to the radial direction. At r = 2(mu), the highest angle that the photon can escape at is (alpha) = 0; at r = 3(mu)...
I am currently learning general relativity and I kind of understand what the symbols in einstein field equationd represent. But I need example like those that involves actual numerical values. I have been trying to search for it online but I cant. So does anyone mind showing me how you apply...