Mathematical Principles of Theoretical Physics

4 CHAPTER 1. GENERAL INTRODUCTION

Here the temporal component of the metric and the gravitational force of the ball exerted on
an object of mass massmare

(1.2.5) g 00 =−

(

1 −

2 MG

c^2 r

)

=−

(

1 +

2

c^2

ψ

)

, F=−m∇ψ=−

GMm

r^2

New law of gravity (Ma and Wang,2014e)

Gravity is the dominant interaction governing the motion and structure of the large scale
astronomical objects and the Universe. The Einstein law of gravity has been a tremendous
success when it received many experimental and observational supports, mainly in a scale of
the solar system.
Dark matter and dark energy are two great mysteries in the scale of galaxies and beyond
(Riess and et al., 1989 ;Perlmutter and et al., 1999 ;Zwicky, 1937 ;Rubin and Ford, 1970 ).
The gravitational effects are observed and are not accounted for in the Einstein gravitational
field equations. Consequently, seeking for solutions of these two great mysteries requires
a more fundamental level of examination for the law of gravity, and has been the main in-
spiration for numerous attempts to alter the Einstein gravitational field equations. Most of
these attempts, if not all, focus on altering the Einstein-Hilbert action with fine tunings, and
therefore are phenomenological. These attempts can be summarized into two groups: (a)
f(R)theories, and (b) scalar field theories, which are all based on artificially modifying the
Einstein-Hilbert action.
Our key observation is that due to the presence of dark matterand dark energy, the
energy-momentum tensorTμ νof visible baryonic matter in the Universe may not conserved:
∇μTμ ν 6 = 0 .which is a contradiction to the Einstein field equations (1.2.3), since the left-
hand side of the (1.2.3) is conserved:∇μ

[

Rμ ν−^12 gμ νR

]

= 0 .The direct consequence of this
observation is to take the variation of the Einstein-Hilbert action under energy momentum
conservation constraint:

(1.2.6)(δLEH(gμ ν),X) =

d

dλ

∣

∣

∣

λ= 0

LEH(gμ ν+λXμ ν) = 0 ∀X={Xμ ν}with∇μXμ ν= 0.

The div-free condition,∇μXμ ν=0, imposed on the variation element represents energy-
momentum conservation. We call this variational principle, the principle of interaction dy-
namics (PID), which will be discussed hereafter again.
Using PID (1.2.6),^1 we then derive the new gravitational field equations (Ma and Wang,
2014e):

Rμ ν−

1

2

gμ νR=−

8 πG

c^4

(1.2.7) Tμ ν−∇μ∇νφ,

∇μ

[

8 πG

c^4

Tμ ν+∇μ∇νφ

]

(1.2.8) = 0.

(^1) The new field equations (1.2.7) can also be equivalently derived using the orthogonal decomposition theorem,
reminiscent to the Helmholtz decomposition, so that
Tμ ν=T ̃μ ν− c
4
8 πG
∇μΦν.
See Chapter 4 and (Ma and Wang,2014e) for details.