Page 155 - Intro to Tensor Calculus

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so that κ (n) must be a root of the determinant equation

det(b αβ − κ (n) a αβ )= 0. (1.5.122)

The expanded form of equation (1.5.122) can be written as

b
2
αβ
κ (n) − a b αβ κ (n) + =0 (1.5.123)
a
where a = a 11 a 22 − a 12a 21 and b = b 11 b 22 − b 12b 21 . Using the definition given in equation (1.5.107) and using
the result from equation (1.5.110), the equation (1.5.123) can be expressed in the form

κ 2 (n) − 2Hκ (n) + K =0. (1.5.124)

The roots κ (1) and κ (2) of the equation (1.5.124) then satisfy the relations

1
H = (κ (1) + κ (2) ) (1.5.125)
2

and
K = κ (1) κ (2) . (1.5.126)
Here H is the mean value of the principal curvatures and K is the Gaussian or total curvature which is the
product of the principal curvatures. It is readily verified that

Eg − 2fF + eG eg − f 2
H = and K =
2
2(EG − F ) EG − F 2
are invariants obtained from the surface metric and curvature tensor.

Relativity
Sir Isaac Newton and Albert Einstein viewed the world differently when it came to describing gravity and
the motion of the planets. In this brief introduction to relativity we will compare the Newtonian equations
with the relativistic equations in describing planetary motion. We begin with an examination of Newtonian
systems.
Newton’s viewpoint of planetary motion is a multiple bodied problem, but for simplicity we consider
only a two body problem, say the sun and some planet where the motion takes place in a plane. Newton’s
law of gravitation states that two masses m and M are attracted toward each other with a force of magnitude
GmM
2 ,where G is a constant, ρ is the distance between the masses, m is the mass of the planet and M is the
ρ
mass of the sun. One can construct an x, y plane containing the two masses with the origin located at the
center of mass of the sun. Let b e ρ =cos φ b e 1 +sin φ b e 2 denote a unit vector at the origin of this coordinate
system and pointing in the direction of the mass m. The vector force of attraction of mass M on mass m is
given by the relation
−GmM
~
F = b e ρ . (1.5.127)
ρ 2