EINSTEIN EQUATION: ALTERNATIVE FORM

Link to: physicspages home page.To leave a comment or report an error, please use the auxiliary blog.Reference: Moore, Thomas A., A General Relativity Workbook, Univer-

sity Science Books (2013) - Chapter 21; Box 21.3.The general relativistic generalization of Newton’s law of gravity is

Gi j +Λgi j = κT i j (1)

where the Einstein tensor is defined in terms of the Ricci tensor and thecurvature scalar as

Gi j ≡ Ri j − 12

gi jR (2)

We can write this in a different form that is sometimes easier to use incalculations. Eliminating Gi j we have

Ri j − 12

gi jR+Λgi j = κT i j (3)

Multiplying both sides by gi j we get

gi jRi j − 12

gi jgi jR+Λgi jgi j = κgi jT i j (4)

Because the tensor gi j is the inverse of gi j, their product gives the identitymatrix of rank 4 (this can be seen by doing the calculation in a local inertialframe where gi j = ηi j and noting that since it’s a tensor equation, it’s validin all coordinate systems). That is

gi jg jk = δki (5)

so if we contract the δ kj tensor we just sum up its diagonal elements and

since these are all 1 (and there are four rows), we get

δkk = 4 (6)

Returning to 4 we get

gi jRi j −2R+4Λ = κgi jT i j (7)1

EINSTEIN EQUATION: ALTERNATIVE FORM 2

Since the curvature scalar is given by

R ≡ gi jRi j (8)

and the stress-energy scalar is

T ≡ gi jT i j (9)

we get

−R+4Λ = κT (10)Multiplying this by −1

2gi j and subtracting from 3 we have

Ri j −Λgi j = κ

(T i j − 1

2gi jT

)(11)

Isolating the Ricci tensor gives us the alternative form of the Einsteinequation:

Ri j = κ

(T i j − 1

2gi jT

)+Λgi j (12)

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