B5 - General Relativity

rip me

the jacobian of a set of curvilinear coords

how to calculate Lagrangian for ds²

divide by 2 x d(tau)²

invariant measure definition

How to calculate Christoffel symbols (Big Gamma) ?

Compute lagrangian, then use ELEs wrt tau. The variable used in the ELE is the top indices, and the bottom two are composed of the terms of dxidxj as equated from the geodisk eq

Covariant derivative of scalar and vector

Covariant derivative of a scalar is the same as a regular derivative

Covariant derivative of a tensor

Christoffel symbol in terms of metric

Riemann tensor symmetries

Ricci tensor def and properties

symmetric

Bianchi identity

Prove from Bianchi identities

Gravitational redshift formula

Z = Lemit - Lobvs / Lemit (L = lambda). Z = R(t0)/R(t) -1

Einstein field eqs

Trace the einstein eqs to find R

Schwarzschild metric

Kretschmann invariant (Big K) and features

A singularity in K is a true singularity

How to find orbital eq

Equate -c²dTau² = metric (usually schwartzschild)

Orbit eq in terms of u = 1/r

Effective pot for Schwartzschild and its minimum

Derive eq of motion for circular orbit

J definition

Friedmann equations

Hubble Paramater

Hubble constant

Equations of state for friedmann equations

w = 0 for dust, w=1/3 for radiation

Friedmann continuity eq

Comoving distance

z and its relation to a and lambda

When can proper time be used as an affine parameter?

When parameterising a time like geodisk

why, in einstein’s eq, is Rmunu not proportional to Tmunu ?

Due to cons of energy, del(Tmunu) = 0 but del(Rmunu) = non 0

what is special about massless particles’ real time?

dtau = 0