Transcript Document

Relativity and Black Holes
Frank Hsia-San Shu
National Tsing Hua University
8 October 2005
Potential Physics & Chemistry Majors
Outline of Lecture
• The speed of light in a vacuum, c = 300,000 km/s, is the
same no matter how fast an observer moves with respect to
the source of the light (special relativity).
– As a consequence, time appears dilated and lengths seem to suffer
(Lorentz) contraction for moving objects.
– As a further consequence, E  mc 2 , the most famous equation in
all of science.
• Locally, it is not possible to distinguish between
gravitation and acceleration (general relativity).
– As a consequence, it is possible for gravity to bend light and to
warp spacetime. Indeed, gravitation is warped spacetime.
– Black holes represent regions where spacetime is so warped that
fundamental challenges are posed to our understanding of physics.
Einstein at 15: Racing at the Speed of
Light Violates Maxwell’s Equations
Theory of Special Relativity:
Two Basic Postulates
• Valid laws of physics are the same for all
inertial observers (people who move at
constant velocity, maybe zero, relative to
the “fixed stars”).
• The speed of light, c = 300,000 km/s, is the
same for all observers, independent of their
motion relative to the source of light.
Time Dilation
Proof
•
One tick of light clock according to woman:
•
One tick of light clock according to man:
d  ctwoman
 ctman
d
•
vtman
Relationship between time intervals:
ctman   d 2  (vtman )2  (ctwoman )2  (vtman )2 .
Solving for t
, we get
woman
twoman  tman
i.e., time passed for woman is factor
v2
1 2 ,
c
1  v 2 / c2
shorter than time passed for man.
Lorentz Contraction
L  L0 1 
v2
.
c2
Twin paradox: Can’t woman regard herself to be at rest?
Won’t she deduce that her brother ages less quickly than she does?
Relativistic Increase in Mass
Elasticcollisionwith equal y - momentum,extracomponentof velocityvx
m*
m
y
mvy  MVy
M*
M
•
•
•
MovingobserverA *  vx
Stationary observer A
M and m have identical mass m0 when they are at rest with respect to each other.
Distance y seen by A and A* is unaffected by motion of A* in x-direction at
velocity chosen to equal vx , which is why vx *  0.
Time t * for collision measured by A* is shorter than t measured by A by a
factor of 1  vx 2 / c 2 . This implies
y y
vy* 
t *

t
 vy.
•
But by momentum balance and symmetry: mv y  MVy  m * v y *.
•
Thus, m  m*  M by the inverse factor
•
The reason is that m as seen by A has an extra component of velocity vx in the
x-direction compared to M seen by the same observer A.
In the limit when vy and Vy go to zero, M and m *go to the rest mass m0, and vx
goes to v, the total velocity. Therefore, we conclude that the mass m of a ball in
1
motion at speed v is larger than its rest mass m0 by the factor  
 1.
1
1  vx / c
2
2
.
1 v2 / c2
Equivalence of Mass and Energy
•
•
According to Newton, what a moving ball has extra compared to a stationary
ball of mass m0 is
1
kinetic energy = m0v 2 .
2
According to Einstein, what a moving ball has extra compared to a nonmoving
1
ball of mass m0 is more mass:
m   m0 where  
•
•
•
Perhaps energy and mass are the same thing!
1 v / c
2
2
 1.
E  mc 2 .
In the above equation, E = total energy. For v << c,
v2
1
  1  2 and E  m0c 2  m0v 2
2c
2 2
2
where the term m0c is called the rest energy and m0v / 2 is the kinetic energy
as defined by Newton. More generally, KE  E  m0c2  (  1)m0c2 .
According to Newton, when a ball is placed in a gravitational field (or more
generally, any attractive field of force), what it has is (negative) potential energy.
For Einstein, what it has is less mass. According to Lavoisier when hydrogen
combines with oxygen to make water, mass is conserved but heat is released.
No, heat is energy, and the water actually contains slightly less mass than the
original hydrogen and oxygen.
Possibility of Converting Energy
into Mass and Vice-Versa
m   m0
m
•
•
•
v
v
m*  m0 *
m
m*
m*
Energy of putty balls before collision = 2mc  2 m0c .
Energy of putty balls after collision = 2m * c2  2m0 * c2.
Energy is conserved in process
 2 m0c2  2m0 * c2.
2
2
 m0*   m0  m0 since   1.
•
•
•
Rest mass after collision is greater than rest mass before collision! There has
been conversion of energy into mass. (Not more putty molecules, but putty
weighs slightly more than it did at rest.)
Conversion of mass into energy must also be possible.
Basis of nuclear power. Although less well known, also basis of chemical
power; indeed, basis of all power.
Newton’s Greatest Achievement,
Einstein’s Happiest Thought
• Apple (and everything else) falls
to Earth at g = 9.8 m/s2 (Galileo).
• Moon “falls” about Earth with
centripetal acceleration (Huygens)
v2/r = 0.0029 m/s2 = g/3600.
• Moon is 60 times farther away
from center of Earth than apple
(Erastothenes). 602 = 3600.
• Maybe gravitation  1 / r 2 .
• Then 2nd and 3rd laws imply
GMm
 ma
2
r
GM
ag 2
r
F
independent of any property of m.
• Nothing can travel faster than
the speed of light. How then
can
GMm
F
?
2
r
• Neither F = ma nor F =
GMm/r2 are relativistically
correct. What can replace these
equations?
• Maybe gravitation and
acceleration are
indistinguishable locally.
• Accept Galileo’s empirical
finding, but not Newton’s
theoretical explanation.
Two Views of Gravitation
• Newton:
– Gravity is a force which pulls on all things with mass.
– Mass acts as the source that generates the force of
gravitation.
• Einstein:
– There is no such thing as the force of gravity.
Gravitation arises when spacetime has curvature;
indeed gravitation is spacetime curvature.
– Mass-energy and stress (e.g., pressure) act as the
sources that generate spacetime curvature.
Theory of General Relativity:
Two Basic Postulates
• The concepts of special relativity apply to local
phenomena.
• Locally, there is no way to distinguish physically
between gravitation and acceleration.
Bending of Light
Gravitational Bending of Light
Ant Analogy for Bending of Light
Lensing of Background Galaxies by
Galaxy Cluster
Event Horizon of Black Hole
• Laplace:
1 2 GMm
mv 
 0.
2
R
For v  c,
2GM
R 2 .
c
• Schwarzschild:

1

for r  R
0
1 R / r
2GM
where R  2 .
c
Flight Circles about a Black Hole
•
•
•
•
•
•
•
•
•
RSch = 9 km for a 3 solar-mass BH.
Start with flight circle of circumference = 2π·90 km.
You deduce you’re 90 km radially from BH. Don’t jump to conclusions.
Lower yourself inward radially by 32 km.
Fly around; measure circumference = 2π·60 km. (?)
Lower yourself inward radially by another 33.75 km.
Fly around; measure circumference = 2π·30 km. (??)
Lower yourself inward radially by another 19.8 km.
You compute 32+33.75+19.8 = 85.55. Subtracted from 90, won’t this
bring you inside RSch = 9 km? (!)
• Don’t worry; lower yourself by 19.8 km as we requested.
• Fly around; measure circumference = 2π·15 km. Whew!
• All flight circles are in a single plane. Clearly, presence of a 3 solarmass point-mass at center has warped our usual (Euclidean) sense of
geometry.
Black Holes Are
Punctures in Fabric of Spacetime
Behavior Near Event Horizon
Reversal of Space and Time Across
Event Horizon of a Black Hole
• Outside event horizon, by exerting enough force on the rope, I can
hold your position stationary with respect to center of BH. But there is
nothing I can do to stop the forward progression of time for you (or,
for that matter, for myself).
• As I lower you toward event horizon, your perception of stars begin to
change and blur. Are you getting a sinking feeling?
• When you get close enough to the event horizon, no rope – no matter
how strong – can stand the strain. It will snap and break, and you will
begin an inexorable fall toward the black hole.
• For you, it takes only a few milliseconds for you to reach and cross
the event horizon of the BH. But for me, it seems that you formally
take an eternity to reach the event horizon.
• In other words, as you draw near to the event horizon, there is nothing
I can do to stop your forward progression through space. But for me,
time seems to have stopped moving for you! In some sense, for me on
the outside, time and space seem to reverse roles as you approach the
event horizon. When you cross it, you will reach a different space and
time than the one that we on the outside occupy. Some people
speculate that BHs may be portals to other spacetimes and other
universes!
Optical Jet Emanating from Nucleus
of M87, an Elliptical Galaxy
Formation of Magnetized Black Hole in
Self-similar Gravitational Collapse
Cai & Shu (2005)
Speculation 1-- Wormholes:
Shortcuts through Space?
y
x
Speculation 2 -- Wormholes:
Machines through Time?
x
t
Speculation 3: Evaporation of BHs?
•
•
•
Currently popular theoretical view:
Proton is a long-lived, but
ultimately unstable particle, which
will decay into positron plus other
particles in some 1032 years or so.
Even BHs may ultimately
evaporate completely away.
According to Hawking,
(nonrotating) BHs of a mass M
have a nonzero surface temperature
T given by the formula:
c 3
kT 
.
8GM
This formula has been recently
been recovered by the methods of
superstring theory, which is the
attempt to reconcile general
relativity with quantum mechanics.
ct
x
Heisenberg uncertainty principle:
Et ~ .
ct ~ 2RSch

c 3
E ~
~ 2kT .
4GM
c 4GM
~
E
c2
c 3
kT ~
.
8GM

Physics as a Community
An unbroken thread from the beginning of
modern science to the present age.
?
Thank you everyone!