Gravitation

ds is the length along a path connecting two nearby points

The role of the metric is that it allows us to define at each point of the manifold the “length” of, and for a Cartesian one the angles between,
the infinitesimal displacement vectors dx^u

the flow of the uth component of momentum through the surface x^v constant (cdV_vp^u)

The equation holds without gravitation (i.e. it reproduces Special Relativity when the metric is the Minkowski metric) and the equation is generally covariant (i.e. it preserves its form under arbitrary x to x' transformations)

The metric is a global stretch of the Minkowskian metric, which means all light cones have 45 degree angles

A curve along which a tangent vector when parallel transported to another point along the curve is parallel to the tangent vector at this new point

A manifold endowed with a metric that is locally Cartesian

A manifold endowed with a metric that is locally Minkowskian

In cosmic space time there exists a set of fundamental observers whose world-lines form a smooth bundle (a “congruence”) of time-like geodesics. This bundle diverges from a point in the past and/or converges to a point in the future. The worldliness don’t intersect, except possibly at the (at most two) singular points described before.

A fluid with no viscosity and heat conduction but particles interact causing a pressure. In the co-moving frame, the fluid is assumed to be isotropic

There are no local experiments which can distinguish non-rotating free fall in a gravitational field from uniform motion in space in the absence of a gravitational field (gravitational mass equals inertial mass)

A spherically symmetric vacuum solution exterior to a mass distribution can always be made static by a suitable choice of variables, and the metric must have the form of the Schwarzschild solution

a frame of reference in which there is no acceleration

The commutator of the covariant derivative applied to a contravariant vector

Determines the rate of expansion of the universe. Sdot/S, where -S^2 is the prefactor of the space comppnent of the line element squared

57Fe emits gamma ray of exactly 14keV. The iron was placed in a loudspeaker at the bottom and then top of a buidling and a scintillation counter was put on the other side of another Fe sample at the other end of the building. A mylar bag of helium was put in the middle to minimise scattering. The speaker was set to vibrate so that the source had a time-varying Doppler shift and gravitational redshift could be measured by looking at the effect at maximum Doppler shift.

a process of transporting a vector along a curve in a way that the vector remains in the same direction physically even though its components will change as the basis vector change

a phenomenon where the gravity of a massive object (such as a galaxy or a cluster of galaxies) bends and distorts the light coming from a distant object behind it

Globally isotropic n-dimensional spaces (n > 2) are manifolds of constant curvature K. The Riemann tensor for an isotropic space is given by
Ruvab = K(guagvb − gubgva) , where K is constant on every point of the manifold.

light emitted by a star that is detected by an observer far away is redshifted by the gravitational field of the star

light emitted by a star is bent around the sun and so appears in a different position. Can compare this to when the sun is not in the way (solar eclipse)

General relativity predicts elliptical orbits with slowly precessing axes and one can measure the change in angle of the perihelions

light travelling through a curved space time takes longer than one would expect if the space time were flat so bounce signals off planets and time then

universe has a beginning and then expands, predicts a Hubble constant linked to the rate of expansion and some background radiation from the beginning event

a tensor whose representation is the same in any coordinate system

the energy density currently that would imply our current universe if flat

Guv = Ruv -1/2guvR = 8piG_N/c^4Tuv

a path of shortest distance between 2 points on a Riemannian manifold

the parameter u is chosen such that the rate of change of u exactly matches the length of the tangent vector in the geodesic equation. Therefore, the only change to the tangent is due to its parallel transport along the curve

At each epoch the universe is homogeneous and isotropic

a is a measure of the relative size of the universe at different times and its derivative represents the rate of change of the size of the universe (r = a(t)r0)

A frame linearly accelerated relative to an inertial frame in special relativity is locally identical to a frame at rest in a gravitational field

the boundary beyond which events cannot be observed from an observer's reference frame

half the affine connection minus itself with the bottom two indicies swapped

v_i' = e_i' . e_jv_j

time for an observer in an inertial frame where they are static at the origin