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Geodesics
---------
Consider a vector field V^{n}. It is contravariant because we are considering
displacement. The rate of change of x with respect to the proper time, τ,
along a world line is given by:
dV^{n}/dτ = (∂V^{n}/∂x^{m})(dx^{m}/dτ)
where dx^{m}/dτ is the TANGENT VECTOR to the world line. It is a
contravariant 4-velocity vector. dτ = √g_{μν}dx^{μ}dx^{ν}
D_{s}V^{n} = D_{m}V^{n}dx^{m}/dτ --- replace with the covariant derivative
= (∂V^{n}/∂x^{m})(dx^{m}/dτ) + Γ^{n}_{mr}V^{r}dx^{m}/dτ
= dV^{n}/dτ + Γ^{n}_{mr}V^{r}dx^{m}/dτ --- chain rule.
How does tangent vector vary along τ? Compute its covariant derivative.
V^{n} -> dx^{n}/dτ. Thus,
D_{s}dx^{n}/dτ = d^{2}x^{n}/dτ^{2} + Γ^{n}_{mr}(dx^{m}/dτ)(dx^{r}/dτ)
The above equation describes the trajectories followed by freely falling
objects in spacetime. If we define these as curves of zero proper
acceleration we can write:
d^{2}x^{n}/dτ^{2} + Γ^{n}_{mr}(dx^{m}/dτ)(dx^{r}/dτ) = 0
Therefore,
d^{2}x^{n}/dτ^{2} = -Γ^{n}_{mr}(dx^{m}/dτ)(dx^{r}/dτ)
This is the equation for a GEODESIC. The geodesic is the equation
of motion that defines the straightest, and therefore shortest, path
through curved space-time. By this we mean that once the
gravitational field is given (i.e. Γ is known) this equation
tells us how the object will move in such a field.
Action of a Geodesic - Particle Moving in Arbitrary Gravitational Field
-----------------------------------------------------------------------
By definition, the action of a particle is -mc^{2}∫dτ
dτ^{2} = (1 + 2φ/c^{2})dt^{2} - (1/c^{2})dx^{2}
A = -mc^{2}∫√[(1 + 2φ/c^{2})dt^{2} - (1/c^{2})dx^{2}]
= -mc^{2}∫√[(1 + 2φ/c^{2}) - (1/c^{2})dx^{2}/dt^{2}]dt
.
= -mc^{2}∫√[(1 + (1/c^{2})(2φ - x^{2})]dt
Use the binomial theorem √(1 + x) = 1 + x/2
.
A = -mc^{2}∫(1 + (1/2c^{2})(2φ - x^{2})dt
So the Lagrangian is:
L = -mc^{2}[(1 + (1/2c^{2})(2φ - x^{2})]
.
= -mc^{2} - mφ + mx^{2}/2