Euler-Lagrange equation  
 
  christoffel-schwarzschild  
 
  ricci-schwarzschild  
 
  Constants  
 
  Units   
 
  home  
 
  ask us  
 

 

      General Relativity



© The scientific sentence. 2010

Relativity: Einstein's equations




1. Energy conservation equation /h3>

We already have:

Gμν = 8 πGTμν

Einstein tried to link the Energy momentum tensor Tμν to the curvature of space. Therefore, the left hand Gμν of the above equation must contain a tensor curvature. This tensor could not be the one of Riemann because it contains three indexes. The right one is then the one of Ricci.

The first attempt was to set Rμν = K Tμν, where K is a constant. But the conservation of the tensor Energy written as:

μTμν = 0

or better, with the covariant derivative: ∇μTμν = 0
will imply ∇μRμν = 0 which is not true.

In fact

μ Rμν = (1/2) gμνμR

R is the curvature scalar, so we can replace the ordinary derivative ∂μR by a covariant derivative ∇μR. Therefore:

μ Rμν = (1/2) gμνμR

We have:
μ (gμν R) = R ∇μ gμν + gμνμR =

Since
μ gμν = 0

Then ∇μ (gμν R) = gμνμR

Hence ∇μ Rμν = (1/2) ∇μ(gμν R)

That is: ∇μ[Rμν - (1/2)(gμν R)] = 0

We set

Gμν = Rμν - (1/2)(gμν R)

Called Einstein's tensor,

and write: ∇μGμν = 0

Energy conservation Equation
μGμν = 0

We have then

Rμν - (1/2)(gμν R) = K Tμν
Equation that expresses the conservation of energy.


In vacuum, where there is no matter,
Einstein's equation is Gμν = 0, because Tμν = 0
.

So
Rμν - (1/2)(gμν R) = 0

Contracting with the metric gμν gives:
gμν [Rμν - (1/2)(gμν R)] = 0

That is:
R - (1/2)(gμνgμν R) = 0
gμνgμν R = δμν R = δμμ R = (1 + 1 + 1 + 1)R = 4 R

Therefore:
R - 2R = - R = 0
That is R = 0

Where there is no matter, there is no curvature.


2. Einstein Equation

Let's solve for the constant K in the equation:
Rμν - (1/2)(gμν R) = K Tμν

Multiplying by the metric gμν gives:
gμν[Rμν - (1/2)(gμν R)] = K gμνTμν

We obtain:
- R = K Tμν = K T

Therefore:
R = - K T, and
Rμν + (1/2)(gμν K T) = K Tμν
or
Rμν = [Tμν - (1/2)(gμν T] K

Recall:

Gμν = 8 πGTμν

Using the Eienstein tensor:

Gμν = Rμν - (1/2)(gμν R)

We have then the value of the constant K
K = 8 πG

Therefore:

Gμν = Rμν - (1/2)(gμν R) = 8 πGTμν

Gμν = Rμν - (1/2)(gμνR) = 8 πGTμν


3. Complete Einstein Equation

The Energy conservation Equation :

μGμν = 0

Is integrated as:
Gμν = constant.

That is :
Rμν - (1/2)(gμνR) = 8 πGTμν + Const.

This equation can be written as:
Rμν - (1/2)(gμνR) + Const = 8 πGTμν

The constant Const is set as equal to Λgμν. Λ is called
the cosmological constant

Therefore the complete Einstein's equation is:

Gμν + gμνΛ = Rμν - (1/2)(gμνR) + gμνΛ = 8 πGTμν

Einstein's Equation:

Rμν - (1/2)(gμνR) + Λgμν = 8 πGTμν


The cosmological constant Λ can be set as follows:

In free space without graviation, Tμν = 0

Contracting :
Rμν - (1/2)(gμνR) + Λgμν = 8 πGTμν
with the metric gμν gives:
gμνRμν - (1/2)(gμνgμνR) + Λgμνgμν = 0

That is:
R - 2 R + 4 Λ = 0

Therefore
R - 2 R + 4 Λ = 0
Λ = R/4

Λ = R/4

Einstein's Equation takes the form:

Rμν - (1/4)(gμνR) = 8 πGTμν








  


chimie labs
|
Physics and Measurements
|
Probability & Statistics
|
Combinatorics - Probability
|
Chimie
|
Optics
|
contact
|


© Scientificsentence 2010. All rights reserved.