Mathematical Principles of Theoretical Physics

2.3. EINSTEIN’S THEORY OF GENERAL RELATIVITY 69

Remark 2.30.The terms in (2.3.38) are

Rμ ν−

1

2

gμ νR=δLR, LRas in (2.3.36),

Tμ ν=δLs, Ls=

∫

M

gμ νSμ ν

√

−gdx.

In Section3.3, we shall give the derivation ofδLR.

Einstein derived the field equations (2.3.38) in 1915 using his great physical intuition.
According to the classical gravity theory, the Newton potentialφsatisfies the Laplace equa-
tion

(2.3.39) ∆φ= 4 πGρ,

whereρis the mass density. Einstein thinks the gravitational fieldequations of general rela-
tivity should be in the form

(2.3.40) Gμ ν=βTμ ν, βis a constant,

whereTμ νrepresents the energy-momentum tensor, which are providedby physical obser-
vations, andGμ νrepresents gravitational potential which are the generalization of∆φin
(2.3.39). HenceGμ νcontains the derivatives ofgμ νup to the second order. In Riemannian
geometry, only the curvature tensors satisfy the needed properties. ThusGμ νmust be in the
form
Gμ ν=Rμ ν+λ 1 gμ νR+λ 2 gμ ν.

whereλ 1 ,λ 2 are constants.
By the conservation of energy-momentum:

∇μTμ ν= 0 ,

the second-order tensorGμ νshould be divergence-free, i.e.

(2.3.41) ∇μGμ ν= 0.

By the Bianchi identity,Gμ νsatisfying (2.3.41) are uniquely determined in the form up to a
constantλ,

(2.3.42) Gμ ν=Rμ ν−

1

2

gμ νR+λgμ ν,

whereλgμ νare divergence-free due to (2.3.28). Furthermore, Einstein determined the con-
stantβin (2.3.40) by comparing with (2.3.39), andβis given by

β=−

8 πG

c^4

.

Thus, by (2.3.40) and (2.4.42), Einstein deduced the field equations in the general form as
follows

(2.3.43) Rμ ν−

1

2

gμ νR+λgμ ν=−

8 πG

c^4

Tμ ν,