Mathematical Principles of Theoretical Physics

1.2 Law of Gravity, Dark Matter and Dark Energy.

1.2 Law of Gravity, Dark Matter and Dark Energy

Gravity is one of the four fundamental interactions/forcesof Nature, and is certainly the first
interaction/force that people studied over centuries, dating back to Aristotle (4th century BC),
to Galileo (late 16th century and early 17th century), to Johannes Kepler (mid 17th century),
to Isaac Newton (late 17th century), and to Albert Einstein (1915).

Newtonian gravity

Newton’s universal law of gravity states that the gravitational force between two massive
objects withmandMis given by

(1.2.1) F=−

GmM

r^2

,

which is an empirical law.

2.3 Einstein’s Theory of General Relativity

One of the greatest discovery in the history of science is Albert Einstein’s general theory
of relativity (Einstein, 1915 , 1916 ). He derives the law of gravity, his gravitational field equa-
tions by postulating two revolutionary fundamental principles: the principle of equivalence
(PE) and the principle of general relativity (PGR):

1) PE says that the space-time is a 4-dimensionalRiemannian manifold{M,gμ ν}

with the Riemannian metric{gμ ν}representing the gravitational potential;

2) PGR says that the law of gravity is covariant under general coordinate

transformations of both the inertial and non-inertial reference frames;

3) PGR, together with simplicity principle of law of Nature, uniquely dictates

the Lagrangian action, also called the Einstein-Hilbert functional:

(1.2.2) LEH({gμ ν}) =

∫

M

(

R+

8 πG

c^4

S

)

√

−gdx;

4) The Einstein gravitational field equations are then derivedusing the least

action principle, also called the principle of Lagrangian dynamics (PLD):

(1.2.3) Rμ ν−

1

2

gμ νR=−

8 πG

c^4

Tμ ν.

This is the most profound theory of science. The PGR is a symmetry principle, and
the law of gravity, represented as a set of differential equations (1.2.3), is dictated by this
profound and simple looking symmetry principle. The connection to the Newtonian gravi-
tational law (1.2.1) is achieved through the following Schwarzschild solutionof the Einstein
field equations in the exterior of a ball of spherically symmetric matter field with massM:

(1.2.4) ds^2 =−

(

1 −

2 MG

c^2 r

)

c^2 dt^2 +

(

1 −

2 MG

c^2 r

)− 1

dr^2 +r^2 dθ^2 +r^2 sin^2 θdφ^2.