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Tensor Analysis Applied to General Relativity
TAM 611 Class Notes by Professor Richard Rand
CORRECTED VERSION
Dept. Theoretical and Applied Mechanics, Cornell University
- Geodesics
Let the line element in some space be given by
ds^2 = gij dxidxj^ (1)
The geodesic curve xi^ = xi(s) is defined by the calculus of variations problem:
δ
∫ (^2)
1
ds = 0 (2)
To simplify the computation we use the trick of noting that, from (1),
ds^2 ds^2
= gij
dxi ds
dxj ds
so that (2) can be written
δ
∫ (^2)
1
1 · ds = δ
∫ (^2)
1
gij
dxi ds
dxj ds
ds = 0 (4)
The Euler equations corresponding to eq.(4) are:
d ds
∂F
∂ x˙k^
∂F
∂xk^
where F = gij x˙i^ x˙j^ and where dots represent differentiation with respect to s. Eq.(5) becomes
d ds
( 2 gik x˙i
) − ∂gij ∂xk^
x˙i^ x˙j^ = 0 (6)
Expanding the derivative term and dividing by 2 we get
gik x¨i^ +
∂gik ∂xj^
x˙j^ x˙i^ −
∂gij ∂xk^
x˙i^ x˙j^ = 0 (7)
If we interchange the i and j dummy indices in the middle term we get
∂gik ∂xj^
x˙j^ x˙i^ =
∂gjk ∂xi^
x˙i^ x˙j^ (8)
so that eq.(7) becomes
gik x¨i^ +
( ∂gik ∂xj^
∂gjk ∂xi
) x ˙i^ x˙j^ −
∂gij ∂xk^
x˙i^ x˙j^ = 0 (9)
Now multiply by gkm^ to get
x¨m^ +
gkm
( ∂gik ∂xj^
∂gjk ∂xi^
∂gij ∂xk
) x ˙i^ x˙j^ = 0 (10)
which may be written
¨xm^ +
{ m i j
} x ˙i^ x˙j^ = 0 (11)
- General Relativity
Reference: “The Meaning of Relativity” by Albert Einstein
We consider the problem of a particle moving in a 1/r^2 gravity field, e.g. a planet moving around the sun. Newton’s equations can be written
EQUATION A: F = −∇V = ma (12)
where the potential energy V = −k/r satisfies Laplace’s equation:
EQUATION B: ∇^2 V = 0 (13)
Einstein replaced equations A and B by other statements which were entirely geometrical. Equa- tion A was replaced by the condition that the particle move on a geodesic in the 4 dimensional space-time continuum. Equation B was replaced by conditions which specified the metric tensor gij of the space-time continuum. We will look at both of these conditions next.
In special relativity the line element ds of the space-time continuum is given by:
ds^2 = dt^2 −
dx^2 c^2
dy^2 c^2
dz^2 c^2
which may be written ds^2 = (dx^4 )^2 − (dx^1 )^2 − (dx^2 )^2 − (dx^3 )^2 (15)
where x^4 = t, x^1 = x/c, x^2 = y/c, x^3 = z/c, and where c is the speed of light. This metric is related to the Lorentz transformation of special relativity and will not be discussed here. It represents “flat” (or “Minkowskian”) space-time. For the curved space-time continuum of general relativity, eq.(15) is generalized to ds^2 = gij dxidxj^ (16)
where i and j go from 1 to 4. Due to the symmetry of the metric tensor, there are 10 independent gij ’s.
Einstein replaces equation A (12) with the statement that in free space a particle moves on a geodesic in space-time:
¨xm^ +
{ m i j
} x ˙i^ x˙j^ = 0 (17)
- Newton’s equations as a first approximation
In order to see the relation between Newton’s equations and general relativity, we need to examine the general relativistic equations (11) and (23) in the limit that: i) c >> 1, and ii) gij is nearly Minkowskian. Assumption ii) gives us that
ds^2 ≈ dt^2 −
dx^2 c^2
dy^2 c^2
dz^2 c^2
= (dx^4 )^2 − (dx^1 )^2 − (dx^2 )^2 − (dx^3 )^2
whereupon assumption i) gives us that
ds ≈ dt = dx^4
The geodesic equation (11) becomes simplified because, by assumption i), ˙x^1 , ˙x^2 and ˙x^3 are all small compared to ˙x^4 ≈ 1:
x¨m^ +
{ m 4 4
} (1)(1) = 0 (24)
where (compare eqs.(10) and (11)) { m 4 4
}
gkm
( ∂g 4 k ∂x^4
∂g 4 k ∂x^4
∂g 44 ∂xk
) (25)
Now we make the further assumption that iii) the field is static, so that the gij do not depend on time t = x^4. This makes the partial derivatives with respect to x^4 vanish in eq.(25), giving { m 4 4
} = −
gkm^
∂g 44 ∂xk^
∂g 44 ∂xm^
where the last approximation is due to assumption ii). Thus Einstein’s geodesic equation of motion (11) becomes
x¨m^ +
∂g 44 ∂xm^
which agrees with Newton’s equation A (12) if we identify
g 44 =
2 V
m
Next let us consider Einstein’s field equations (23):
Rmijm =
∂xj
{ m i m
} −
∂xm
{ m i j
}
{ r i m
} { m r j
} −
{ r i j
} { m r m
} = 0 (29)
Assumption ii) leads to the conclusion that the last two terms are small of second order:
Rmijm ≈
∂xj
{ m i m
} −
∂xm
{ m i j
} = 0 (30)
Consider in particular eq.(30) for i = j = 4:
Rm 44 m ≈
∂x^4
{ m 4 m
} −
∂xm
{ m 4 4
} = 0 (31)
The first term in eq.(31) is zero by assumption iii) (static field), giving
∂ ∂x^1
{ 1 4 4
}
∂x^2
{ 2 4 4
}
∂x^3
{ 3 4 4
} = 0 (32)
Using eq.(26) in eq.(32), we obtain ∇^2 g 44 = 0 (33)
which with eq.(28) gives Newton’s equation B (13), ∇^2 V = 0.
- Precession of Mercury
In this section we omit the assumptions i),ii) and iii) of the previous section. Schwarzschild in 1916 found an exact solution of the field equations in spherical coordinates r, φ (=longitude), and θ (=colatitude):
ds^2 =
( 1 −
R
r
) dt^2 −
1 − R r
dr^2 c^2
− r^2
dθ^2 c^2
− r^2 sin^2 θ
dφ^2 c^2
where R is a constant of integration. (Incidentally, the singularity at r = R in eq.(34) is associ- ated with “black holes”.)
Using the gij of eq.(34) in the geodesic equation (11), and setting u = 1/r and θ=π/2 (for planar motion), there results the following equation on the orbit of a particle moving around the sun:
d^2 u dφ^2
Rc^2 2 h^2
Ru^2 (35)
where h is a constant of integration. The first three terms of eq.(35) are the usual Newtonian equations. The corresponding linear ODE is a simple harmonic oscillator and has the solution
u = A + B cos φ ⇒ r =
A + B cos φ
which is the equation of an ellipse. The effect of general relativity is to add the fourth term of eq.(35). The ODE is now nonlinear but can be solved. The resulting solution predicts that the ellipse (36) will precess, in agreement with observations made on the planet Mercury.
