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April Uhr Exercise 1: Minkowski space (7 Points) In special relativity time and euclidian three-dimensional space are unified in a four- dimensional vector space called Minkowski space

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1. Übungsblatt zur Vorlesung SS 2016

Allgemeine Relativitätstheorie Prof. G. Hiller

Abgabe: bis Dienstag, den 26. April 2016 16:00 Uhr

Exercise 1: Minkowski space (7 Points)

In special relativity time and euclidian three-dimensional space are unified in a four- dimensional vector space called Minkowski space. Spacetime events are described by contravariant4-vectors

¡xµ¢

=

x0 x1 x2 x3

= µc t

~x

= µt

~x

, (1)

wheret is the time coordinate,~xis the position vector andc=1denotes the speed of light.

Additionally,covariant4-vectors are defined as

xµ=X

ν ηµνxνηµνxν where ηµν=

−1, µ=ν=0 +1, µ=ν=1, 2, 3

0, otherwise

. (2)

We employ theEinstein summation convention: Whenever an index appears twice, sum- mation over this index is implied.

The tensorηµν is themetricof Minkowski space; it can be used to map contravariant vectors onto covariant vectors and vice versa (it “lowers” and “raises” indices). Its inverse ηµνis defined byηµνηνρ=δµρ.

(a) Calculate or simplify (explicitly in terms of the componentsx0,x1,x2,x3) the fol- lowing expressions:

(i) xν=ηµνxµ (ii) ηλλ=ηµνηµν (iii) ηαβηγβ (iv) ηµνxνxµ

(v) ηµαxσησαxµ

(b) The scalar product of two 4-vectorsxµandyµis given by the expressionxµyµ. Linear transformationsΛµ0µthat map 4-vectors onto a new set of coordinates (labeled by µ0) and leave the scalar productxµyµinvariant are calledLorentz transformations.

Contravariant 4-vectors transform according to

xµ0µ0νxν. (3)

Derive the respective transformation law for covariant 4-vectors. How does the derivativeµxµ transform? (Hint: Chain rule)

1

(2)

Exercise 2: Tensor properties (6 Points) LetSµν =Sνµ be a symmetric tensor; Aµν= −Aνµ an antisymmetric one. Let Tµν be an additional arbitrary tensor of rank 2. Arbitrary tensorsTµ1µ2...µn of rankn can be symmetrized or antisymmetrized according to

T(µ1µ2...µn):= 1 n!

X

P

Tµ1µ2...µn (4)

and

T[µ1µ2...µn]:= 1 n!

X

P

sgn(P)Tµ1µ2...µn, (5)

respectively. Here,Pdenotes the permutations of the indicesµiandsgn(P)is the sign of the permutation, defined bysgn(P)=

½ +1, ifP is even

−1, ifP is odd . (a) Show explicitly:

SµνTµν=SµνT(µν), AµνTµν=AµνT[µν], SµνAµν=0. (6) (b) Show that an arbitrary rank-2 tensor can be decomposed into a symmetric and an

antisymmetric part:

Tµν=T(µν)+T[µν]. (7) Can this also be done for tensors of rankn>2? Provide a proof or a counterexample.

(c) Show that in general

Tµν6=Tνµ. (8)

Exercise 3: Light cone, proper time and 4-velocity (7 Points) The norm of a 4-vectorxµ is defined as

x2xµxµ=ηµνxµxν. (9) (a) Describe the physical meaning of the following three cases. Sketch your results in

an|~x|–t-diagram.

(i) xµxµ=0 (ii) xµxµ<0 (iii) xµxµ>0

For trajectoriesxµ(λ)that obeyxµxµ<0we can define theirproper timeτby

2= −ηµνd xµd xν. (10) (b) Show that the differential proper timeis related to the differential coordinate timed t via=d t/γand derive an explicit expression forγ. Which values can1/γ take?

(c) Calculate the norm of the 4-velocityuµd xµ .

2

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