SECTION B Answer ONLY TWO questions in Section B [Expect to use one single-sided A4 page for each Section-B sub question.] Question B1 Consider the line element where w is a constant. ds²=-dt²+e2wt dx², a) Determine the components of the metric and of the inverse metric. [2 marks] b) Determine the Christoffel symbols. [See the Appendix of this document.] [10 marks] c) Write down the geodesic equations. [5 marks] d) Show that e2wt it is a constant of geodesic motion. [4 marks] e) Solve the geodesic equations for null geodesics. [4 marks] The Schwarzschild line element in the usual coordinates x" = (t, r, 0, ) is a) ds² = - 1- 2GM dr.2 dt² + 2GM +r² (d02+ sin² Odo²). 1 r State whether this spacetime is asymptotically flat, and briefly justify your answer. b) The non-vanishing Christoffel symbols are [4 marks] Ittr = GM r(r - 2GM)' Ir tt = GM(r-2GM) 1-3 GM Irr = − r(r - 2GM)' Tee (r-2GM), 1 = (r-2GM) sin² 0, re= го sin cos 0, r 1 1 ΓΦ TO = Гроф r tan 0 Write down the t-component of the geodesic equations. (The other components are not requested.) c) Show that E = = (1-2GM) t [5 marks] is constant along a geodesic. Supposing that the geodesic is a timelike one, identify the natural parameter of the geodesic with respect to which we take the derivative in t, and explain how that parameter is defined. State what is the physical meaning of the constancy of ε. [8 marks] d) The super-hero Hulk kicks a super-tough ball upwards so hard that the ball escapes the Earth's gravitational field. Explain why the ball's trajectory must obey ε > 1 in order for the ball to never fall back down, and determine the final velocity of the ball as a function of ε. Ignore air resistance. [8 marks]
SECTION B Answer ONLY TWO questions in Section B [Expect to use one single-sided A4 page for each Section-B sub question.] Question B1 Consider the line element where w is a constant. ds²=-dt²+e2wt dx², a) Determine the components of the metric and of the inverse metric. [2 marks] b) Determine the Christoffel symbols. [See the Appendix of this document.] [10 marks] c) Write down the geodesic equations. [5 marks] d) Show that e2wt it is a constant of geodesic motion. [4 marks] e) Solve the geodesic equations for null geodesics. [4 marks] The Schwarzschild line element in the usual coordinates x" = (t, r, 0, ) is a) ds² = - 1- 2GM dr.2 dt² + 2GM +r² (d02+ sin² Odo²). 1 r State whether this spacetime is asymptotically flat, and briefly justify your answer. b) The non-vanishing Christoffel symbols are [4 marks] Ittr = GM r(r - 2GM)' Ir tt = GM(r-2GM) 1-3 GM Irr = − r(r - 2GM)' Tee (r-2GM), 1 = (r-2GM) sin² 0, re= го sin cos 0, r 1 1 ΓΦ TO = Гроф r tan 0 Write down the t-component of the geodesic equations. (The other components are not requested.) c) Show that E = = (1-2GM) t [5 marks] is constant along a geodesic. Supposing that the geodesic is a timelike one, identify the natural parameter of the geodesic with respect to which we take the derivative in t, and explain how that parameter is defined. State what is the physical meaning of the constancy of ε. [8 marks] d) The super-hero Hulk kicks a super-tough ball upwards so hard that the ball escapes the Earth's gravitational field. Explain why the ball's trajectory must obey ε > 1 in order for the ball to never fall back down, and determine the final velocity of the ball as a function of ε. Ignore air resistance. [8 marks]
University Physics Volume 1
18th Edition
ISBN:9781938168277
Author:William Moebs, Samuel J. Ling, Jeff Sanny
Publisher:William Moebs, Samuel J. Ling, Jeff Sanny
Chapter1: Units And Measurement
Section: Chapter Questions
Problem 54P: Suppose [V]=L3,[]=ML3, and [t]=T . (a) What is the dimension of dV? (b) What is the dimension of...
Related questions
Question
![SECTION B
Answer ONLY TWO questions in Section B
[Expect to use one single-sided A4 page for each Section-B sub question.]
Question B1
Consider the line element
where w is a constant.
ds²=-dt²+e2wt dx²,
a) Determine the components of the metric and of the inverse metric.
[2 marks]
b) Determine the Christoffel symbols. [See the Appendix of this document.]
[10 marks]
c)
Write down the geodesic equations.
[5 marks]
d) Show that e2wt it is a constant of geodesic motion.
[4 marks]
e)
Solve the geodesic equations for null geodesics.
[4 marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c0d52ac-ebeb-4643-8051-a3a6486fdce4%2Fcc5e9c05-8107-4f52-89f8-a58ba6ed8c87%2Fz1x4h6h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:SECTION B
Answer ONLY TWO questions in Section B
[Expect to use one single-sided A4 page for each Section-B sub question.]
Question B1
Consider the line element
where w is a constant.
ds²=-dt²+e2wt dx²,
a) Determine the components of the metric and of the inverse metric.
[2 marks]
b) Determine the Christoffel symbols. [See the Appendix of this document.]
[10 marks]
c)
Write down the geodesic equations.
[5 marks]
d) Show that e2wt it is a constant of geodesic motion.
[4 marks]
e)
Solve the geodesic equations for null geodesics.
[4 marks]
![The Schwarzschild line element in the usual coordinates x" =
(t, r, 0, ) is
a)
ds² = - 1-
2GM
dr.2
dt² +
2GM
+r² (d02+ sin² Odo²).
1
r
State whether this spacetime is asymptotically flat, and briefly justify your answer.
b) The non-vanishing Christoffel symbols are
[4 marks]
Ittr =
GM
r(r - 2GM)'
Ir tt =
GM(r-2GM)
1-3
GM
Irr = −
r(r
- 2GM)'
Tee (r-2GM),
1
=
(r-2GM) sin² 0,
re=
го
sin cos 0,
r
1
1
ΓΦ
TO
=
Гроф
r
tan 0
Write down the t-component of the geodesic equations. (The other components are not
requested.)
c)
Show that
E =
= (1-2GM)
t
[5 marks]
is constant along a geodesic. Supposing that the geodesic is a timelike one, identify the
natural parameter of the geodesic with respect to which we take the derivative in t, and
explain how that parameter is defined. State what is the physical meaning of the constancy
of ε.
[8 marks]
d) The super-hero Hulk kicks a super-tough ball upwards so hard that the ball escapes the
Earth's gravitational field. Explain why the ball's trajectory must obey ε > 1 in order for the
ball to never fall back down, and determine the final velocity of the ball as a function of ε.
Ignore air resistance.
[8 marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6c0d52ac-ebeb-4643-8051-a3a6486fdce4%2Fcc5e9c05-8107-4f52-89f8-a58ba6ed8c87%2Fspb2pz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The Schwarzschild line element in the usual coordinates x" =
(t, r, 0, ) is
a)
ds² = - 1-
2GM
dr.2
dt² +
2GM
+r² (d02+ sin² Odo²).
1
r
State whether this spacetime is asymptotically flat, and briefly justify your answer.
b) The non-vanishing Christoffel symbols are
[4 marks]
Ittr =
GM
r(r - 2GM)'
Ir tt =
GM(r-2GM)
1-3
GM
Irr = −
r(r
- 2GM)'
Tee (r-2GM),
1
=
(r-2GM) sin² 0,
re=
го
sin cos 0,
r
1
1
ΓΦ
TO
=
Гроф
r
tan 0
Write down the t-component of the geodesic equations. (The other components are not
requested.)
c)
Show that
E =
= (1-2GM)
t
[5 marks]
is constant along a geodesic. Supposing that the geodesic is a timelike one, identify the
natural parameter of the geodesic with respect to which we take the derivative in t, and
explain how that parameter is defined. State what is the physical meaning of the constancy
of ε.
[8 marks]
d) The super-hero Hulk kicks a super-tough ball upwards so hard that the ball escapes the
Earth's gravitational field. Explain why the ball's trajectory must obey ε > 1 in order for the
ball to never fall back down, and determine the final velocity of the ball as a function of ε.
Ignore air resistance.
[8 marks]
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