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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 <code> g''ab x^a x^b </code> sum''i sum''j g''ij x''i x''j
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,
dz'''dy, dy'''dx, 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 <code> int_[[t</code>a, 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 <code> x, p </code> dx/dt

@L/@x = -@/@x V(x)
From @L/@p <code> m (dx/dt), d/dt( @L/@p) </code> 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( g''ab d''x^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

What is the basis of GR (q) - Revision history

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