Basic form of Einstein’s field Equation

R↵� �12g↵�R = ��T↵�

“matter tells spacetime how to curve”

Einstein’s field equation has 16 components, so there are 16 equations, but only 10 of them are independent, because the metric and stress-energy tensor is symmetric about the diagonal.

These are regarded as 10 differential equations for the 10 independent metric tensor components, and are very complicated.

There was only 1 solution to this equation for the first 50 years!

PHYM432 Relativity and Cosmology12. Einstein’s Field Equation & Newtonian Limit

Newtonian approximationIn the limit of small velocities (v << c) and weak fields, Einstein’s equations should reproduce Newtonian gravity. This satisfies the third founding principle of GR, and is how Einstein figured out what the constant was.��T↵�

Assuming the field will be weak means that space will almost be flat, and we can approximate the metric coefficients with

g↵� ⇡ �↵� + h↵� static weak field metric

where h↵� << 1

so, in 2D:

or with which will simplify things

(dS)2 = (⌘00 + h00)(cdt)2 + (⌘11 + h11)dr

2

g↵� =

(1 + h00) 0

0 �(1� h11)

�d✓ = d� = 0

Newtonian approximation

We can then use the weak field metric with the equations of motion

and Einstein’s equation to recover Newtonian mechanics.

0 = md2X�

d�2+ m��

µ⇥dXµ

d�

dX⇥

d�

R↵� �12g↵�R = ��T↵�

we start with (1). In our simple 2D case, we have 2 equations of motion

(1)

(2)

� = 0

� = 1

0 =

0 =

If we are non-relativistic and at low velocity ( v << c) and many of the terms can be neglected

p0 >> p1

�mc >> �mv

c >> v

So we can ignore all terms of as well as terms.

mdX0

d�>> m

dX1

d�dX1

d�

dX1

d�

dX0

d�

0 = md2X0

d�2+ m�0

00

✓dX0

d�

◆2

0 = md2X1

d�2+ m�1

00

✓dX0

d�

◆2

� = 0

� = 1

Our two equations of motion are then

Newtonian approximation

�000 =

�µ�� = � 1

2gµµ

�g��

�XµFor a symmetric metric:

0 = md2X0

d�2+ m�0

00

✓dX0

d�

◆2The first equation of motion becomes

� = 0

0 = md2X0

d�2

mdX0

d�= constant

p

0 = constant = E/c

So the 1st equation states that energy is conserved

Newtonian approximation

0 static solution - no time dependance

The 2nd of motion becomes

�100 =

0 = md2X1

d�2+ m�1

00

✓dX0

d�

◆2

� = 1

0 = md2X1

d�2+ m

✓12rh00

◆ ✓dX0

d�

◆2

0 = md2r

dt2+ mc2 1

2rh00

Newtonian approximation symmetric metric�↵

�� = � 12g↵↵

✓�g��

�X↵

◆�1

00 = � 12g11

✓�g00

�X1

◆

�100 = � 1

2(�11 + h11)

✓⇥(�00 + h00)

⇥X1

◆= +

12(1� h11)

✓⇥h00

⇥X1

◆

keep only leading order�100 =

12(1 + h11 + h2

11 + ...)✓

�h00

�X1

◆

�100 ⇡

12

✓�h00

�X1

◆⇡ 1

2rh00

expand

1st equation

m

✓dX0

d�

◆=

E

cd(ct)d�

=E

mc1d�

=1dt

E

mc2

✓1d�

◆2

=✓

1dt

◆2 ✓E

mc2

◆2

0 = md2X1

d�2+ m

✓12rh00

◆ ✓E

mc

◆2

0 = mdr2

dt2

✓E

mc

◆2 1c2

+ m

✓12rh00

◆ ✓E

mc

◆2

useful relations

0 = md2r

dt2+ mc2 1

2rh00

If we relate h00 =

2�

c2

If we get

0 = md2r

dt2+ mcr�

ma = �mGM/r2

F = �mGM

r2So we recover newton’s law of gravity

� = �GM

rand

The weak field metric is then fully given by

Newtonian approximation

(dS)2 =✓

1 +2�

c

2

◆(cdt)2 �

✓1� 2�

c

2

◆[dx

2 + dy

2 + dz

2]

Next we need to solve the limiting case for Einstein’s field equation

R↵� �12g↵�R = ��T↵�

With a simple stress-energy tensor (Dust) we have

T↵� = T00 = �c2

so the R00 component is all we need.

R00 =

R0000 =

Newtonian approximation

(2)

R0000 + R1

001

R0000 =

��000

�x0� ��0

00

�x0+ �0

00�000 + �1

00�010 � �0

00�000 � �1

00�010

0R0000 =

R

�↵�� =

@��↵�

@x

��

@��↵�

@x

�+

X

�

��↵���

�� �X

�

��↵���

��

R1001 =

R00 = R1001 = �1

2r2h00

So the Ricci tensor is

Newtonian approximation

R1001 =

��101

�x0� ��1

00

�x1+ �0

01�100 + �1

01�110 � �0

00�101 � �1

00�111

�100 ⇡

12rh00 �1

11 ⇡12rh11

Order h2 so we neglect last termR1

001 ⇡ �12r2h00

R1001 ⇡ ���1

00

�x1

R

�↵�� =

@��↵�

@x

��

@��↵�

@x

�+

X

�

��↵���

�� �X

�

��↵���

��

Einstein’s equation can be re-written as (exercise 4.7 in the book)

R↵� = ��(T↵� �12g↵�T )

So in our weak field approximation we have

R↵� =

�12r2h00 = ��

12⇥c2

So for Einstein’s field equation we have

r2h00 = �⇥c2

r22⇤/c2 = �⇥c2comparing with Newton Poisson equation

Newtonian approximation

contract T00

only 1 term

R00 = �⇥

✓T00 �

12(�00 + h00)T )

◆

�12r2h00 = �⇥

✓T00 �

12(�00)⇤c2

◆

R00 = ��(T00 �12g00T )

�12r2h00 = ��

✓⇥c2 � 1

2⇥c2

◆

neglect h00

T = T 00 = �c2

r22⇤/c2 = �⇥c2comparing

with Newton Poisson equation r2⇤ = 4�G⇥

� =8⇥G

c4The constant is

So we now know the full form of the Field equation

G↵� = �8�G

c4T↵�

R↵� �12g↵�R = �8�G

c4T↵�

Newtonian approximation

Originally, Einstein thought the Geodesic equation was an additional requirement to complete GR, but it was later shown to be predicted from the field equation itself, as r↵G↵� = 0 r↵T↵� = 0

So Einstein’s field equation predicts it’s own equation of motion!

So Einstein’s field equation reducing to the Newtonian Poisson equation completes the proof that Newtonian gravity is the limiting case of GR.

The static weak field metric fully describes Newtonian mechanics with the first leading order of the equations, and the effects of GR in this regime can be seen as a small correction factor h↵�

Most astronomical systems are well described by Newton’s laws, and the equivalent weak field metric for GR in the first order.But there are two cases in the solar system where the effects are important, and the higher orders we neglected are needed. These post-Newtonian effects (as their called) became the first famous tests of GR.

They are1) Precession of the perihelion of Mercury2) Bending of light by the sun

Lambourne

4.7