Mathematical Principles of Theoretical Physics

4.2 Physical Supports to PID

It implies that the usual energy-momentum tensor satisfies

(4.2.3) ∇μTμ ν= 0.

However, due to the presence of dark matter and dark energy, the energy-momentum
tensor of visible matterTμ νmay no longer be conserved, i.e. (4.2.3) is not true. Hence we
have
∇μTμ ν 6 = 0 ,

which is a contradiction to (4.2.1) and (4.2.2).
On the other hand, by the Orthogonal Decomposition Theorem3.17,Tμ νcan be orthog-
onally decomposed into

(4.2.4) Tμ ν=T ̃μ ν−

c^4

8 πG

∇μΦν,

andT ̃μ νis divergence-free:

(4.2.5) ∇μT ̃μ ν= 0.

Hence, by (4.2.2) and (4.2.5) the gravitational field equations (4.2.1) should be in the form

(4.2.6) Rμ ν−

1

2

gμ νR=−

8 πG

c^4

T ̃μ ν.

By (4.2.4) we have

T ̃μ ν=Tμ ν+ c

4

8 πG

∇μΦν,

which stands for all energy and momentum including the visible and the invisible matter and
energy, and which, by (4.2.5), is conserved. Thus, the equations (4.2.6) are rewritten as

(4.2.7) Rμ ν−

1

2

gμ νR=−

8 πG

c^4

Tμ ν−∇μΦν,

The equations (4.2.7) are just the variational equations ofLEHwith the div-free constraint as
(4.1.33). Namely, (4.2.7) are the gravitational field equations obeying PID.
We remark that the term∇μΦνin (4.2.7) has no variational structure, and cannot be
derived by modifying the Einstein-Hilbert functional. Hence, (4.2.7) are just the variational
equations due to PID.

4.2.2 Non well-posedness of Einstein field equations

The second strong theoretical evidence to PID is that the classical Einstein gravitational field
equations are an over-determined system.
The Einstein field equations (4.2.1) possess 10 unknown functionsgμ νand 10 indepen-
dent equations due to symmetry for the indicesμandν. However, since the general coordi-
nate system can be arbitrarily chosen, under proper coordinate transformation

̃xμ=aνμxν, 0 ≤ν≤ 3 ,