Redshift Academy


Einstein's Field Equations
--------------------------

We will first state the Einstein field equations and then walk through 
the general rationale that Einstein used to derive them (without getting
into the enormously complicated mathematics).

The equations are:

G_μν = 8πGT_μν

Where,

G_μν = R_μν - (1/2)g_μνR 

Where,

R_μν is the RICCI tensor.
g_μν is the METRIC tensor.
R is the SCALAR CURVATURE.  R = g^μνR_μν.  This corresponds to the trace 
  of the Ricci curvature tensor.
T_μν is the STRESS-ENERGY tensor.

The Einstein equation actually consists of 16 separate equations. In a 
nutshell, they show how energy and momentum (aka mass) determine the 
curvature of spacetime.  Energy and mass are represented on the right 
hand side of the equations while curvature is represented on the left 
hand side.

The Riemann tensor can be expressed entirely in terms of the Christoffel 
symbols and their first partial derivatives.  In turn, the Christoffel 
symbols are comprised of the partial derivatives of the metric tensor. 
Therefore, the Riemann tensor is comprised of the first and second 
derivatives of the metric tensor.  Thus, for the Rindler metric we 
would have:

        -                 -
g_μν =  | (1 + 2φ)  0  0  0 |
     |    0     -1  0  0 |
     |    0      0 -1  0 |
     |    0      0  0 -1 |
        -                 -

R_μνσ^λ = ∂_νΓ^λ_μσ - ∂_μΓ^λ_νσ + Γ^ρ_μσΓ^λ_ρν - Γ^ρ_νσΓ^λ_ρμ

Where,

Γ^a_bc = (1/2)g^ad{∂g_dc/∂x^b + ∂g_db/∂x^c - ∂g_bc/∂x^d}

    = (1/2)g^ad{∂_bg_dc + ∂_cg_db - ∂_dg_bc}

In Newtonian mechanics, a freely falling object accelerating towards 
the earth implies a force acting between the object and the earth 
corresponding to gravity.  In general relativity, however, the earth 
and the object are considered to be both moving along geodesics 
(along straight lines with zero proper acceleration) and it is the 
acceleration produced by the convergence of these geodesics as 
a result of the curvature of the spacetime that we observed as the
object falls.  For example, imagine 2 ships starting a certain distance 
apart at the equator of the earth and sailing due north along different 
lines of longitude (geodesics).  As they proceed, their east west 
separation diminishes until they finally meet at the north pole.  We 
can imagine this 'coming together' as being the result of a force 
(acceleration) with both transverse and longitudinal components
acting on both objects. Thus,



Mathematially, if the separation 4-vector between the earth and 
the object at the same proper time, τ, as both advance along 
their respective geodesics is ξ^μ, the coordinate acceleration 
between them is given by the GEODESIC DEVIATION EQUATION (not 
proven here but fairly straightforward to work out): 

D²ξ^μ/Dτ² = -R^μ_νρσ(dx^ν/dτ)(dx^ρ/dτ)ξ^σ

Where D is the covariant derivative and R^μ_νρσ is the RIEMANN tensor. 

The geodesic deviation equation assigns TIDAL forces to the Riemann 
tensor.  The equation yields a component of acceleration along the 
geodesic (the Γⁿ_mr(dx^m/dτ)(dx^r/dτ) term in the original geodesic 
equation) and components transverse to the direction of motion.  
Together these accelerations act together to produce tidal forces 
on the objects.  The components along the geodesic result in
elongation while the tranvserse components result in compression.

Another way to see this is to imagine an object in flat spacetime 
moving towards a region of curvature.  When the object encounters 
the region it must 'adjust' dimensionally in the transverse and 
longitudinal (radial) directions in order to follow the metric of 
the curved spacetime at each point.  In the process of adjusting, 
the object experiences non-uniform forces and becomes stretched
in the direction of motion and squeezed in transverse direction.  

Uniform gravitational fields that are 'simulated' by accelerated
reference frames do not have curvature.  Such fields do not exist 
in nature (real gravitational fields involve mass and curvature). 
However, the gravitational field near the surface of a large object 
over a relatively small solid angle can be considered to be 
approximately uniform.  Since they do not have curvature, a free 
falling object in a uniform gravitational field will not experience 
any tidal forces.

The Weak Field Approximation
----------------------------

The weak field approximation is analagous to perturbation theory and 
allows simplifying the study of many problems in general relativity
while still producing useful approximate results.  Equations can be 
obtained by assuming the spacetime metric is only slightly different 
from the Minkowski metric, η, where η is constant.  Thus,

g_μν = η_μν + h_μν and g^μν = η^μν - h^μν

The Geodesic Equation
-----------------------

Look at acceleration of a particle along a geodesic.  In the limit
this should reduce to the familiar Newtonian force law.

d²xⁿ/dτ² = -Γⁿ_mr(dx^m/dτ)(dx^r/dτ)

x⁰ = t = τ, m = 1, and dτ² = g_μνdx^μdx^ν, v << c

If the velocity is small compared to c, the spatial components can 
be ignored.  Therefore,

d²x/dt² + Γ^μ₀₀(dx⁰/dτ)(dx⁰/dτ) = 0

d²x/dt² = -Γ^μ₀₀   since dx⁰/dτ = dτ/dτ = 1

From before we can write Γ in terms of the metric,

Γ^a_bc = (1/2)g^ad{∂_bg_dc + ∂_cg_db - ∂_dg_bc}

Γ^a₀₀ = (1/2)(η^ax - h^ax){∂₀h_x0 + ∂₀h_x0 - ∂_xh₀₀}

Since they very small we can neglect all terms that involve products 
of h and its derivatives.  Thus:

Γ^a₀₀ = (1/2)η^ax{∂₀h_x0 + ∂₀h_0x - ∂_xh₀₀}      

    = (1/2)η^ax{2∂₀h_x0 - ∂_xh₀₀}     

    = -(1/2){2∂₀h_x0 - ∂_xh₀₀}     

If the velocity is assumed to be small, the derivatives with respect
to space dominate over the derivatives with respect to time.  
Therefore,

Γ^a₀₀ = (1/2)∂_xh₀₀

Therefore, we can write:

d²x/dt²  = -(1/2)∂_xh₀₀

However, in Newtonian mechanics force F = -∂Φ/∂x (m = 1) where φ 
is the GRAVITATIONAL POTENTIAL ENERGY.  Thus, we can write:

F = -∂Φ/∂x = d²x/dt² 

Substitution yields,

2∂Φ/∂x = ∂h₀₀/∂x

So,

h₀₀ = 2φ + constant

In flat space (φ = 0) h₀₀ = 1 therefore the constant equals 1.  
We end up with:

h₀₀ = g₀₀ = 1 + 2φ

Which is the same form found in the Rindler metric associated with
accelerated reference frames.

dτ² = (1 + 2φ)dt² - dx² 

The Field Equations
------------------

R_μν - (1/2)g_μνR = 8πGT_μν

g^μνR_μν - (1/2)g^μνg_μνR = g^μν8πGT_μν 

R - (1/2)δ^μ_μR = 8πGT

                -       -
            | 1 0 0 0 |
Now δ^μ_μ = δ^ν_ν = | 0 1 0 0 | = 4
            | 0 0 1 0 |
            | 0 0 0 1 |
                -       -

Thus, R is just the trace of R_μν

R - 2R = 8πGT

-R = 8πGT

Substituting this back into the EFEs gives:

R_μν + (1/2)g_μν8πGT = 8πGT_μν

R_μν = 8πG(T_μν - (1/2)g_μνT)

T = T₀₀ + T₁₁ + T₂₂ + T₃₃ ... the trace of T_μν

Consider 00 component with T₀₀ = ρ.  Thus,

R₀₀ = 8πG(ρ - (1/2)g₀₀ρ)

R₀₀ = 8πG(ρ - (1/2)ρ)

R₀₀ = 4πGρ   ... 1.

From before, the Riemann tensor is:

R_μνσ^λ = ∂_νΓ^λ_μσ - ∂_μΓ^λ_νσ + Γ^ρ_μσΓ^λ_ρν - Γ^ρ_νσΓ^λ_ρμ

The ΓΓ terms can be neglected since they are the products of the 
derivatives of h.  Therefore,

R_μνσ^λ = ∂_μΓ^λ_νσ - ∂_νΓ^λ_μσ

Γ^μ_νσ = (1/2)η^μλ(∂_νh_λσ + ∂_σh_λν - ∂_λh_νσ)

Γ^λ_νσ = (1/2)η^λρ(∂_νh_ρσ + ∂_σh_ρν - ∂_ρh_νσ)

Γ^μ_μσ = (1/2)η^λρ(∂_μh_ρσ + ∂_σh_ρμ - ∂_ρh_μσ)

R_μνσ^λ = (1/2)η^λρ(∂_μ∂_νh_ρσ + ∂_μ∂_σh_ρν - ∂_μ∂_ρh_νσ - ∂_ν∂_μh_ρσ - ∂_ν∂_σh_ρμ + ∂_ν∂_ρh_μσ)

R_μνσ^λ = (1/2)η^λρ(∂_μ∂_σh_ρν - ∂_μ∂_ρh_νσ - ∂_ν∂_σh_ρμ + ∂_ν∂_ρh_μσ)

Multiply by η^νρ to get the Ricci tensor:

R_μσ = (1/2)η^νρ(∂_μ∂_σh_ρν - ∂_μ∂_ρh_νσ - ∂_ν∂_σh_ρμ + ∂_ν∂_ρh_μσ)

Multiply by η^σρ to get the Ricci tensor:

R_μν = (1/2)η^σρ(∂_μ∂_σh_ρν - ∂_μ∂_ρh_νσ - ∂_ν∂_σh_ρμ + ∂_ν∂_ρh_μσ)

Pick the 00 component:

R₀₀ = (1/2)η^νρ(∂₀∂₀h_ρν - ∂₀∂_ρh_ν0 - ∂_ν∂₀h_ρ0 + ∂_ν∂_ρh₀₀)

As before, if the velocity is assumed to be small, the derivatives 
with respect to space dominate over the derivatives with respect 
to time.  Therefore,

R₀₀ = (1/2)η^νρ∂_ν∂_ρh₀₀

R₀₀ = -(1/2)∂_i∂_jh₀₀  ... 2.

From 1. and 2. we can write:

-(1/2)∂_i∂_jh₀₀ = 4πGρ

or

∇²h₀₀ = -8πGρ

But h₀₀ = 1 + 2φ.  Therefore,

∇²φ = 4πGρ

Which is the Newtonian equation for the gravitational field.

Einstein's Process for Developing the Field Equations
-----------------------------------------------------

From Newton's Law of Gravity:

∇²φ = 4πGρ

The solution to this equation is the familiar:

φ = -GM/r

and

F = ma = -m∇φ = GmM/r²

Einstein wanted an equation of a similar form.  Geometry and curvature 
on the LHS, Energy density/momentum (the source of the gravitational 
field) on the RHS.

From the weak field theory g₀₀ = 1 + 2φ

∴ φ = (1/2)g₀₀ - 1/2

∴ (1/2)∇²φ = 4πGρ

∴ ∇²g₀₀ = 8πGρ

∴ ∇²g₀₀ = 8πGT₀₀

This was Einstein's first clue that somehow mass effects the curvature 
of space.   Now, the energy-momentum tensor has 2 indeces, is symmetric 
and its divergence equals 0.  Thus,

T_μν = T_νμ and D_μT_μν = 0

So the object representing the curvature on the LHS has to be a tensor 
with the same properties.  The obvious choice was the Ricci tensor but
while it is symmetric, it does not have a divergence equal to 0.  Einstein
started with the Riemann tensor and used the second BIANCHI IDENTITIES 
in conjunction of the asymmetric nature of the Riemann tensor, to find
something that did.

The second BIANCHI identities are written as:

D_λR_αβμν + D_νR_αβλμ + D_μR_αβνλ = 0

The Riemann tensor exhibits the following symmetries:

R_αβμν = -R_βαμν = -R_αβνμ = R_μναβ

Note:  R_αβμν = g_δνR_αβ^δ_μ

Also, we can make use of the fact that Dg = 0 so we can move g inside 
or outside of the covariant derivatives at will.

If we multiply the above Bianchi identity by g^αμ we get:

g^αμD_λR_αβμν + g^αμD_νR_αβλμ + g^αμD_μR_αβνλ = 0

Now g^αμR_αβλμ = -g^αμR_αβμλ= -R_βλ

So we get:

D_λR_βν - D_νR_βλ + D_μR^μ_βνλ = 0

If we multiply the above by g^βν we get:

g^βνD_λR_βν - g^βνD_νR_βλ + g^βνD_μR^μ_βνλ = 0

∴ D_λR - D^βR_βλ + D_μ^βR^μν_βνλ = 0

∴ D_λR - D^βR_βλ + D_μ^βR^μν_βνλ = 0

Interchange μ and ν and from the asymmetry properties we get:

D_λR - D^βR_βλ - D_ν^βR^νμ_βμλ = 0

∴ D_λR - D^βR_βλ - D^βR_βλ = 0

∴ D_λR - 2D^βR_βλ = 0

Alternatively,

2D^βR_βλ - D_λR = 0

∴ D^βR_βλ - (1/2)D_λR = 0

∴ D^βR_βλ - (1/2)g_βλD^βR = 0

∴ D^β[R_βλ - (1/2)g_βλR] = 0

G_βλ = R_βλ - (1/2)g_βλR

G_βλ is both symmetric and has a divergence of 0.  Einstein could 
now write:

G_βλ = 8πGρT_βλ 

Einstein - Hilbert Derivation
-----------------------------

The Einstein field equations can also be obtained very elegantly 
through the principle of least action.  Consider an small area of 
the gravitational potential energy field, φ, in curved spacetime.

            √(g_xx)dx
            ----
  √(g_yy)dy |  φ  |
         |     |
            ----

Area = √(g_xxg_yy)dxdy

This is a special case of a metric with only diagonal components.
In reality the metric will also have g_xy and g_yx components so we can 
generalize by writing:

dA = √(|g|)dxdy where |g| is the determinant of g (i.e. g_xxg_yy - g_xyg_yx)

We can further generalize this to a small volume element in curved
spacetime by writing:

dV = √(-|g_μν|)d⁴x

The minus sign arises because the time component is negative.

Therefore,

V = ∫√(-|g_μν|)d⁴x

This integrand has the form of a Lagrangian density which is an invariant 
quantity.  Now, it is possible to multiply the integrand by a scalar with no
loss of the invariance.  The scalar takes the form of the curvature scalar,
R, plus terms that represent the energy/matter fields, L_M.  Thus we can 
write:

S = ∫√(-|g_μν|)(k'R + L_M)d⁴x

where k' is defined to be c⁴/16πG

After a lot of work that involves starting with the Riemann tensor 
(constructed out of Christoffel symbols and derivatives of Christoffel 
symbols, which are turn constructed out of derivatives of the metric), 
contracting indeces to get the Ricci tensor, contracting again to get the 
curvature scalar, multiplying by the square root of the determinant of the 
metric tensor, factoring in the mass/energy terms and then minimizing the 
action using the Euler-Lagrange equations, one finally obtains the EFEs.

The Cosmological Constant
-------------------------

Einstein recognized there was an ambiguity in his equations.  In order to 
achieve a stationary universe he had to include another term as:

G_μν + Λg_μν = kT_μν

The CC is only a factor on galactic scales since the term is otherwise 
very small and can be ignored.