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This is exercises about chapter 6, Exercises of Relativity Theory

National Taiwan UniversityRelativity Theory

This is exercises about chapter six

Typology: Exercises

2025/2026

Uploaded on 04/05/2026

b12202058
b12202058 🇹🇼

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bg1
6
.
7
(i)
<x(Xy
+
mz)"
=
XY(0
,
(xy)
+
Muz)
+
quixy
+
Mz)
=
X*(x(uY
+
Maz
+
(u(xY
+
MEY)
-
XX
↑
(OY
+
TRY)
+
MX
*
18 E
+
TE)
=
X(Xxy))
+
M(Xxz)
(ii)
M
+
x
+
gyz
=
( fX*
+
gyn)(2xz
TE
=
fXM(2pz
+
TYE)
+
gym(2ZPTEY
=
f(Xxz)"
+
g(Xyz)
(ii)
xx(fY)
=
X
*
(2
,
(fy)
+
TifyY
=
X
*
(1f)4
+
+
&YPfivYVY
=
X
*
(G
,
+)y
+
=
7 X
*
(2,Y
+
TYY)
=
(xf)y
+
f(XxY)e
pf3
pf4
pf5

Partial preview of the text

Download This is exercises about chapter 6 and more Exercises Relativity Theory in PDF only on Docsity!

(i)

  1. (^9) .XM def 5 = (^) 5(5) is monotonic^ and^ differentiable

so

differentiate (1st) , 2

and again (^) A = (2nd) ! = ( = 0

-^ = => A + = (^0) ↑ (^) -

we want this^0 =>

So we^ want & 0

to fit^

the eq = 0 means that (^) (s) = X S + (^) & (a +^0 , a^ , BER)

Id

, B^ are^ constants)

(^6). 20. = (^) Egad(2bgac

. (^) grb-2190c) - (1) Dogan = OcGab-Igab-Tagad from (^) 1) (^) Tca = Egl : geattages

Gegen) & ic = g

(2 (^) , geb

Obgec-Degcb) McGab = OcGab--degab(cGea

Gage-Jega)-

gad(gen-Oge-Gegs Sa = (^25) Goat

  • Ega)=Jagac-ac = (^0)

This

McGab

O

deduce (^) DrXa = (^) GacDoX

Xa= gacX from Pob^ 6- Db(gacX") = Mogac) X^

gacTnX

XbXa = ganMoX a geodesic X

CX) is a^ curve whose tangent vector (^) k =@ satisfies^ KTyk

or

jab

= (^2)

. /Gab

gab

  • (^) on

"Jagab

ir(or) = 29

end

Therefore

, A

but it is proportional to a

Therefore this in the

non-affinely parametrized geodesic^ eq ~ in the^ metric gab Conclus :^ if (^) I is (^) wull

geodesic

of

gab

it is still^ the^ null^ geodesic of Jab =&"gab while (^) it's (^) not affinely parametrized still (^) we can^ change the^ J^ to^ make^ it affinely parametriced . (a = 0)