Introduction to General Relativity. Note 2

Introduction to General Relativity. Note « 2 »

Constantly accelerated elevator
Let the coordinates inside the elevator be (τ,ξ) such that τ is the time inside the elevator and ξ is the distance from the floor. An observer in the outer space uses Cartesian grid (t,x). If the floor of the elevator moves with constant acceleration g the transformation between these two frames is

x = (ξ+1/g) cosh( ^τ/_(ξ+1/g) )-1/g, t = (ξ+1/g) sinh( ^τ/_(ξ+1/g) ).

With new coordinates gη=^τ/_(ξ+1/g) and ρ=ξ+1/g the transformation law simplifies

x = ρ cosh(gη)-1/g, t = ρ sinh(gη).

We notice important effects in the accelerated frame:

The elevator coordinates are curved: ds² = dt² - dx² = (gρ)²dη² - dρ²
The local clock rate in the elevator varies with height: ds²|_dρ=0 = (gρ)²dη²
Equal η lines converge towards the horizons -- the boundaries that separate the part of the total space-time unavailable for the observer in the elevator.

The equivalence principle tells that all these effects must as well be present in gravitational fields.

Curved coordinates (d'Inverno: Chapters 5,6; t'Hooft: Chapters 4,5)
A set of four quantities A^a (a=0,1,2,3) is called a contravariant vector if under a transformation of coordinates x→x'(x) it transforms as coordinate differentials dx^a, that is: A^a = (^{dx^a}/_dx'^b)A'^b.
A set of four quantities A_a (a=0,1,2,3) is called a covariant vector if under a transformation of coordinates x→x'(x) it transforms as derivatives of a scalar ^dφ/_dx^a, that is: A_a = (^{dx'^b}/_dx^a)A'_b.
The scalar product A^aB_a≡∑_aA^aB_a is invariant under coordinate transformation A^aB_a = A'^aB'_a

Exercises

Consider our constantly accelerated elevator with the coordinates (η,ρ) and the reference frame (t,x). With flat coordinates (t,x) the trajectory x(t) of a free moving body satisfies the equation ^d²x/_dt²=0. What are the corresponding equations for the free moving body in the internal (curved) coordinates (η,ρ) ?
Hints:
1. Instead of x(t) consider a parametric description of the curve, (t(s),x(s)), with the corresponding system of equations ^d²x/_ds²=0, ^d²t/_ds²=0.
2. In this system of equations perform a substitution of variables x=ρ cosh(gη)-1/g, t=ρ sinh(gη).
3. Make suitable linear combinations of the equations to reduce them to the canonical (geodesic) form as in d'Inverno, eq.(6.37).
Estimate the difference between the clock rates at the bottom and at the top of the mount Everest in the Great Himalayas.