Relativity and Black HolesFrank Hsia-San Shu

National Tsing Hua University8 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, 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.

2 ,E mc

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:

• Relationship between time intervals:

Solving for , we get

i.e., time passed for woman is factor shorter than time passed for man.

2

21 ,woman man

vt t

c

2 21 /v c

womand c t

manc t d

manv t

2 2 2 2( ) ( ) ( ) .man man woman manc t d v t c t v t

womant

Lorentz Contraction

2

0 21 .

vL L

c

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

• M and m have identical mass when they are at rest with respect to each other.• Distance seen by A and A* is unaffected by motion of A* in x-direction at velocity chosen to equal , which is why • Time for collision measured by A* is shorter than measured by A by a

factor of . This implies

• But by momentum balance and symmetry:

• Thus, by the inverse factor

The reason is that as seen by A has an extra component of velocity in the x-direction compared to seen by the same observer A.

• In the limit when and go to zero, go to the rest mass , and goes to v, the total velocity. Therefore, we conclude that the mass of a ball in motion at speed v is larger than its rest mass by the factor

m

M

y

*M

*m

Stationary observer A

y ymv MV

y

*t t

M

* .*y y

y yv v

t t

* *.y y ymv MV m v

0m

*m m M

m

yv yV M 0mm

0m

and *m

xv * 0.xv

xv *Aobserver Moving

xvy velocity ofcomponent extra momentum,- equalwith collision Elastic

22 /1 cvx

./1

122 cvx

xv

xv

.1/1

122

cv

Equivalence of Mass and Energy

• According to Newton, what a moving ball has extra compared to a stationary ball of mass is

• According to Einstein, what a moving ball has extra compared to a nonmoving ball of mass is more mass:

• Perhaps energy and mass are the same thing!

• In the above equation, E = total energy. For v << c,

where the term is called the rest energy and is the kinetic energy as defined by Newton. More generally,

• 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.

0m 20

1kinetic energy = .

2m v

0m0 2 2

1 where 1.

1 /m m

v c

20m c 2

0 / 2m v2 2

0 0KE ( 1) .E m c m c

2.E mc

20

202

2

2

1 and

21 vmcmE

c

v

Possibility of Converting Energy into Mass and Vice-Versa

• Energy of putty balls before collision = • Energy of putty balls after collision =• Energy is conserved in process

• 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.

0* *m m

v vm m *m *m

2 202 2 .mc m c

2 202 * 2 * .m c m c

2 20 02 2 * .m c m c

0 0 0 * since 1.m m m

0m m

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 • Then 2nd and 3rd laws imply

independent of any property of m.

• Nothing can travel faster than the speed of light. How then can

• 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.

?2r

GMmF

./1 2r

2

2

r

GMga

mar

GMmF

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:

• Schwarzschild:

.2

,For

.02

1

2

2

c

GMR

cvR

GMmmv

.2

where

for /1

1

2

0

c

GMR

RrrR

Flight Circles about a Black Hole• = 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 = 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 solar-

mass point-mass at center has warped our usual (Euclidean) sense of geometry.

SchR

SchR

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 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:

• This formula has been recently been recovered by the methods of superstring theory, which is the attempt to reconcile general relativity with quantum mechanics.

Heisenberg uncertainty principle:

3210

.8

3

GM

ckT

.~ tE

.2~4

~

4

~ 2~

3

2

kTGM

cE

c

GM

E

cRtc Sch

x

ct

.8

~3

GM

ckT

Physics as a Community

An unbroken thread from the beginning of modern science to the present age.

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Thank you everyone!