What is the basis of GR (q)
For people, who are unfamilar with GR, i will cover the overall issue with this quick tutorial
The orgins of GR curved space
In special theory of relativity, space-time interval
ds^2 = (cdt)^2 - (dx^2 + dz^2 +dy^2)
is constant independent of any frame. In his persuits of theory of gravity, Einstien realized that
ds^2 = (cdt)^2 - (dx^2 + dz^2 +dy^2) Equ(1)
was inadequeate to cover gravity. Instead he turned the boarder generalization of this formula
ds^2 gab x^a x^b sumi sumj gij xi xj
Equ(2)
This formula is the heart of curved space geometry. Unlike Equ(1), which has only contributions only from dx^2,dy^2,dz^2, Equ(2) has contribution from dx*dy, dzdy, dydx, etc.
g_ab is called a metric. Roughly speaking, its components are used to compute length of the curve.
In SR, observers disagree on dt,dx,dz,dy components in different frames. However, they all agree on space time interval ds^2. In GR, observers disagree on any of the components of g_ab. However, they all agree with the value of ds^2.
Origins of Einstien's Geodesic Equation
The geodesic equation can be derived from the principle of least action and from the euler lagrange equations. Here's the summary of what these things are, for the curious ones.
The Lagragrain is defined as K.E - P.E. In classical mechanics,
L = 1/2 m (dx/dt) ^2 - V(x)
Action S is defined as
S int_[[ta, t=b]] L dt
According to principle of least action, the particle will travel a path the minimizes L. From the principle, you ger euler-lagrange equations.
@L/@q - d/dt(@L/@p) = 0
Here q is the generalized position coordinate and p is the generalized momentum coordinate
In our classical case, q x, p dx/dt
@L/@x = -@/@x V(x)
From @L/@p m (dx/dt), d/dt( @L/@p) m d^2 x/dt^2
m d^2 x/dt^2 = - @/@x V(x)
This is Netwon's famous 2nd law. This is naturally derived from principle of least action.
Einstien's geodesic equation can be derived by assuming
L = sqrt( gab dx^a /dlamda d_x^b/ dlamda )
where lamda is some form of proper time
In a way, g_ab are actually a measure of gravitional potential and a body travels along a line of least resistances. Moroever, a body travels along a path that minimizes the metric g_ab.
Origins of the Field Equation
Guv = Ruv - (1/2)Rguv
where R is the ricci tensor, R is the radius of curvature and guv our metric.
Although i wouldnt say i fully understand where this comes from, Hilbert derived it also from the principle of least action.
Hilbert's action = sqrt(-g)R
where g is determinant of the metric matrix g_uv. Btw, compare this to volume element of the Jacobian transformations