General relativity

https://physics.stackexchange.com/questions/363/books-for-general-relativity

http://math.ucr.edu/home/baez/gr/gr.html

many people recommend it (e.g. on Reddit)
http://www.preposterousuniverse.com/grnotes/

http://www.physicspages.com/Index%20-%20Relativity.html

• Your Bookmark on Page 43 | Added on Sunday, August 4, 2013 12:00:22 AM
  illustration how a unit cell in c-t space shifts to a parallelogram, not very interesting
  
• Your Bookmark on Page 96 | Added on Sunday, August 4, 2013 12:15:55 AM
  same…
  
• Your Bookmark on page 110 | Added on Sunday, September 27, 2015 4:13:18 PM
  addition of velocities? not interesting…
  

Parallel transport: TODO

Riemann tensor: describes how vector is changed during the parallel transport along a loop. Describes curvature.

R(u, v, w) = R^a_bcd u^b v^c w^d

Ricci tensor:
R_bd = R^c_bcd

R^a_d = g^ab R_bd

Ricci scalar: R^a_a

Geometrically, the Ricci curvature is the mathematical object that controls the growth rate of the volume of metric balls in a manifold.

Paralllel transport on the Earth surface might change the direction

For fun, notice that if we think of the earth as a unit sphere, it's area is 4 pi, and our Roman has just travelled around a region of land having area one eighth of that, hence pi/2. His javelin has rotated by an angle of pi/2! This is no coincidence: on the unit sphere, whenever you go around a simple closed curve enclosing an area A, parallel translation gives a rotation of angle A.


Again, all this is easier to visualize in 2d space rather than 4d spacetime. A person walking on a sphere `following their nose' will trace out a geodesic -- that is, a great circle. Suppose two people stand side-by-side on the equator and start walking north, both following geodesics. Though they start out walking parallel to each other, the distance between them will gradually start to shrink, until finally they bump into each other at the north pole. If they didn't understand the curved geometry of the sphere, they might think a `force' was pulling them together.

Similarly, in general relativity gravity is not really a `force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial. If you toss a ball, it follows a parabolic path. This is far from being a geodesic in space: space is curved by the Earth's gravitational field, but it is certainly not so curved as all that! The point is that while the ball moves a short distance in space, it moves an enormous distance in time, since one second equals about 300,000 kilometers in units where $c = 1$. This allows a slight amount of spacetime curvature to have a noticeable effect.

Problem Book in Relativity and Gravitation, A. Lightman and R. H. Price (Princeton University Press, Princeton, 1975).

http://math.ucr.edu/home/baez/einstein/node15.html


Contravariant vectors describe those quantities where the distance units comes at the numerator (like velocity), whereas covariant are those where the distance unit is at the denominator (like temperature gradient).

Covariant vectors are linear funcitonals.

Spherical coordinates:
ds = dr^2 + r^2 dtheta^2 + r^2 sin^2 theta d phi^2
g_11 = 1
g_22 = r^2
g_33 = r^2 sin^2 theta
Non constant metric tensor!

g^u_v = delta^u_v
g^v_u = delta^v_u


Riemann tensor symmetries and identities https://en.wikibooks.org/wiki/General_Relativity/Riemann_tensor

Парадокс альфы центавры

TODO: https://en.wikipedia.org/wiki/Four-vector
I don't like the word "norm" in the article.


Twin paradox in closed spaces (for instance, toroidal space):

* Compact space case http://arxiv.org/abs/gr-qc/0101014
* http://arxiv.org/abs/astro-ph/0606559
* http://arxiv.org/abs/gr-qc/0503070
Global preferred inertial frame
* Jeffrey R. Weeks: The Twin Paradox in a Closed Universe

TODO: okay, what about curved?

'Compactification' breaks the Lorentz invariance.
* Poincare symmetry

http://van.physics.illinois.edu/qa/listing.php?id=15308

In a closed loop with circumference R, there is a preferred rest frame. Special relativity predicts that someone moving with respect to the loop will measure its circumference as being smaller than R, so the preferred rest frame is one that views the closed loop with its maximum circumference.

https://www.physicsforums.com/threads/twin-paradox-in-a-closed-universe.375432/

gamma m (1, v, 0, 0)

photons:

E (1, 1, 0, 0)

For massive particles: P^\mu = m U^\mu; m is the rest mass

Four-momentum is conserved

Lorentz factor: gamma = 1 / sqrt{1 - v^2}


Scalar curvature: an invariant of a Riemannian manifold

Dark energy: cosmological constant??

https://en.wikipedia.org/wiki/Four-vector

http://math.ucr.edu/home/baez/einstein/

General Relativity by Leonard Susskind. Особенно хороши серии 6 и 9. В 9-й type-directed вывод уравнения Эйнштейна, невероятно красиво.

https://github.com/MITGameLab/OpenRelativity

raytracer on haskell

https://physics.stackexchange.com/questions/47919/why-is-the-covariant-derivative-of-the-metric-tensor-zero

https://math.stackexchange.com/questions/24294/is-there-a-good-way-to-compute-christoffel-symbols

https://math.stackexchange.com/questions/1432985/why-may-geodesic-not-be-the-shortest-path-on-a-surface

The same principle applies to the merging black holes. We have two objects that can't actually be real black holes because in any finite universe we know real black holes cannot exist. However they are experimentally indistinguishable from real black holes.

Alternatively, a super supermassive black hole with the mass of 4.3 billion Suns would have a density equal to one i.e. the same density as water.

For some reason it is popular to ignore the part where you have to wait a long time and you only get approximately the simple solution. Really it is just that there are only a small number of rather simple back holes that things can approach over time. There are many ways it can be as it approaches the final state. It's like if you leave some hot metal out in a room, there is just one final temperature it approaches, an equilibrium temperature, but it takes forever to reach it, so it is always a bit hotter than the room but starts to get really really really close.

n the present day universe, this does not happen for two reasons. First, the gas is unstable to fragmentation as it collapses. The reason for this is that the Jeans mass, the smallest mass that is likely to collapse, scales as T3/2/ρ1/2, where T is the temperature and ρ the density. If the gas can cool as it collapses, then the temperature remains roughly constant, the Jeans mass falls and the cloud breaks up into smaller cores. These cores are usually much smaller than the (at least) several solar masses required to form a black hole (see below).

The thing is, with an extra solar mass of material at just a little more than 3 kilometers away from the predicted central singularity, we now have two solar masses of material in this particular region of space. The event horizon of a two solar mass object is 6 kilometers in radius, meaning that all of this material is now inside the event horizon, after all!

Black holes really do grow over time, and all observers can agree exactly when and by how much.

I guess basically, different observers may disagree how big are the black holes.

For an observer falling into the black hole with a lot of mass, the black hole would be smaller. Whereas for an observer outside the black hole, they might already be inside the 'new' even horizon

DavidSJ 36 days ago [–]

David Hilbert discovered general relativity at practically the same moment as Einstein.

gnramires 26 days ago [–]

I believe there was a race at that time, physicists knew there was a need to generalize SR to non-inertial frames, and that perhaps that it involved metric spaces (following the lorentz transformations). Hilbert was working on it, but apparently did not make much progress at deriving the GR equations (I don't have a source, sorry -- scattered articles on GR). You're right that eventually he or someone else would have made it, but how long would it take is speculation.

Edit: More information here https://en.wikipedia.org/wiki/Relativity_priority_dispute

I think the main takeaway is that Hilbert and Einstein ended in amicable terms, and it seems both contributed greatly to eachother's result.