2. Mercator Projection

Exercise Sheet No. 1

Introduction to General Relativity

Exercise Sheet No. 1

1. Tensor Operations

a) Let space be two dimensional, with coordinate (x¹, x²). Suppose the tensors V^a, W_a, P^ab, Q_ab and M^a_b are measured to have the values

V¹ = 2, V² = 3; W₁ = 4, W₂ = 5 (1) P^ab =

2 −1

3 6

ab

; Qab =

0 2 4 7

ab

; M^ab =

4 3 2 1

ab

. (2) Calculate the following tensors:

(1) α=V^aW_a (2) T^b =P^abWa

(3) F^a_c =P^abQ_bc (4) G_ab =M^c_bQ_ca

b) Translate the following 3-vector identities into index notation, and prove them:

(1) A~·(B~ ×C) =~ B~ ·(C~ ×A) =~ C~ ·(A~×B)~ (2) ∇ ·(f ~A) =A~· ∇f +f∇ ·A~

(3) ∇ ·(A~×B) =~ B~ · ∇ ×A~−A~· ∇ ×B~

The cross product can be calculated using the three-dimensional total-antisymetric Levi-Civita symbol, which is defined as:

_αβγ =







1 if αβγ is an even permutation of (1,2,3)

−1 if αβγ is an odd permutation of (1,2,3) 0 otherwise (two indices are equal)

(3)

Using matrix notation it can be displayed in three dimensions as:

_αβ1 =





0 0 0

0 0 1

0 −1 0



; _αβ2 =





0 0 −1 0 0 0 1 0 0



; _αβ3 =





0 1 0

−1 0 0 0 0 0



 (4) or in two dimensions as:

αβ =

0 1

−1 0

(5)

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Exercise Sheet No. 1

In index notation the cross product can now be expressed as:

(A×B)_α=_αβγA^βB^γ (6)

c) LetS^ab andA^abbe symmetric and anti-symmetric contravariant tensors. Show that Sab and Aab are symmetric and anti-symmetric as well.

d) Lets define the rank 3 tensor anti-symmetric tensor A_abc= 1

2 V_abc−V_cba

(7) And A_abc =−A_cba. Show that T^abA_acb vanishes.

e) Show that T^ab_e =T^abc_deδ^d_e is a tensor.

f) Show that T^ab=−T^ba in one coordinate system it implies thatT^{0 ab} =−T^{0 ba} in all coordinate systems.

2. Mercator Projection

The Mercator projection is a cylindrical projection which aims to align compass bearings to constant directions on the map. In other words, if the angle between between a line of longitude and a great circle that intersect at some point on the Earth’s surface isφ, the Mercator projection’s aim is to ensure that the same angle φ is measured on the map.

Figure 1: A schematic visualization of the Mercator map projection. The angle ψ on the surface of the earth does not change when when mapping to a flat plane.

As in the cylindrical case, we start with the line element.

ds² =R² dθ² + sin²θdφ²

, (8)

On the map however, dy=Rdθ and dx=Rsinθdφ

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Exercise Sheet No. 1

a) Show that that if x= _2π^wφ then y = w

2πlog

cot θ 2

(9)

b) Let us define a new variable ˆ

y ≡ 2π

wy = log

cot θ 2

(10) Show that sinθ = sech ˆy (Hint: use the definition of sech and other trigono- metric identiies).

c) With these choices of (x, y), show that the mercator line elements is:

ds² =

2πR w

2

sech²yˆ dx²+ dy²

, (11)

and the Mercator metric:

g_ab =

2πR w

2

sech²yˆ 1 0

0 1

. (12)

Figure 2: A Mercator projection of the earth from 1569.

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Exercise Sheet No. 1

Figure 3: A Mercator projection of the earth including a measure of the distortion known as the “Tissots indicatrix”

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