RELATIVITY OF SPACE AND TIME IN POPULAR SCIENCE RICHARD ANANTUA x 0 - - PowerPoint PPT Presentation

RELATIVITY OF SPACE AND TIME IN POPULAR SCIENCE

RICHARD ANANTUA x0: 7:30p EST xi: Flatiron Institute, Center for Computational Astrophysics

RELATIVITY OF SPACE AND TIME IN POPULAR SCIENCE: TALK OUTLINE

• 1. Popular Science Jeopardy
• 2. Unpopular Science (lots of tensors):

a) Electromagnetism b) Special Relativity c) General Relativity

• 3. Fate of the Universe
• References:
• Interstellar (2014) - Directed by Christopher Nolan, Interstellar portrays a bleak future of a climate-change-

ravaged Earth and a daring mission led by Cooper’s team to chart out escape plans in a planetary system around the supermassive black hole Gargantua.

• The Time Machine (1895) – Authored by H. G. Wells, this classic tome portrays a gripping class conflict nearly a

million years into the future of Victorian England, where our ancestors split into two unrecognizable classes— with both upper and lower appearing completely devoid of humanity

JEOPARDY

Are We Alone? Time Travel Astronomy Astrology Cosmology or Cosmetology? To Infinity and Beyond! X X X X X X X X X X X X 300 X X X X X X X 400 X X X X 500 X X X X . .

JEOPARDY – TIME TRAVEL

• JEOPARDY – ARE WE ALONE

• JEOPARDY – ASTRONOMY

JEOPARDY – FINAL JEOPARDY

MATH METHODS – VECTOR CALCULUS

• Vectors in 3D may be multiplied using
• Dot product:
• Cross product:
• A vector field F(x,y,z) is specified by assigning a vector to each point in space. At each point we may

calculate

• Divergence (whether the vector field looks like a source or a sink)
• Curl (whether a paddlewheel would rotate in the vector field)

! " ⃗ $ = &$

!

&' + &$

"

&) + &$

#

&*

!× ⃗ $ = &$

!

&' − &$

"

&) &$

#

&) − &$

!

&* &$

"

&* − &$

"

&* a×⃗ , = (.",! − ,".!)0 * + (.!,# − ,!.#)0 ' + (.#," − ,#.") ̂ ) a 3 ⃗ , = .#,# + ."," + .!,! = ., cos 7

z z z

ELECTROMAGNETISM – MAXWELL’S EQUATIONS

• Maxwell’s equations describe the dynamics of electric and magnetic fields

! " + = ,

• $

! " . = 0 !×+ = − &. &2 !×. = 3$⃗ 4 + 3$-$

%& %'

Gauss’s Law Gauss’s Law for Magnetism Faraday’s Law Ampère Law Now that we know these basic relationships between + and . … Let there be light! Poynting flux (power per unit area): ⃗ 5 = 1 3$ +×. Let’s also see Maxwell’s Eqs. in integral form q × ⃗ $! v . Lorentz Force Law: ⃗ $ = 8(+ + v×.)

ELECTROMAGNETISM – MAXWELL’S EQUATIONS

• Rewriting Maxwell’s equations in integral form

Differential Form Integral Form ! " + = ,

• $

! " . = 0 !×+ = − &. &2 !×. = 3$⃗ 4 + 3$-$

%& %'

; + " < ⃗ = = >()*

• $

∯ . " < ⃗ = = 0 @ + " <⃗ A = − &Φ+ &2 @ . " <⃗ A = 3$C()* + 3$-$ &Φ& &2 Gauss’s Law Gauss’s Law for Magnetism Faraday’s Law Ampère Law

SPECIAL THEORY OF RELATIVITY – ON THE ELECTRODYNAMICS OF MOVING BODIES

• In 1905 Albert Einstein (1879-1955) wrote observations from a series of thought experiments

and observations in On the Electrodynamics of Moving Bodies:

• Relative, not absolute motion, establishes current in magnet-conductor system
• Simultaneity depends on an observer’s state of motion, or reference frame
• Clocks synchronize if the difference in times (measured locally) from when light is

emitted at A to when light arrives at B equals the corresponding difference for the reverse journey

• In the frame of the conductor, the moving magnet creates a changing magnetic flux that results in an

electric field in the conductor, and subsequently a current

• In the frame of the magnet…?

@ + " <⃗ A = − &Φ+ &2 S N

RELATIVITY OF E AND B - MAGNET AND CONDUCTOR

I v +

• In the frame of the conductor, the moving magnet creates a changing magnetic flux that results in an

electric field in the conductor, and subsequently a current

• In the frame of the magnet, the moving conductor’s free charges are deflected by the magnetic field

@ + " <⃗ A = − &Φ+ &2 ⃗ $! = 8v×. S N

RELATIVITY OF E AND B - MAGNET AND CONDUCTOR

I I v v S N ⃗ $! + B, .

POSTULATES OF SPECIAL RELATIVITY

• Postulate 1: “The laws by which the states of physical systems undergo change are not affected,

whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion” (Einstein, 1905, p.4).

• Postulate 2: “Any ray of light moves in the ‘stationary’ system of co-ordinates with the determined

velocity c, whether the ray be emitted by a stationary or by a moving body” (Einstein, 1905, p.4). The laws of physics are the same in any two inerWal frames The speed of light in a vacuum is the constant c regardless of the moWon of the emiXng frame

E = 299,792,458 m/s

REFERENCE FRAMES

• Frame S is a set of space and time coordinates with Origin O at (0,0,0,0).
• At other points r in S, clocks at rest in S are synchronized with the origin by accounting for light travel

time r/c (simultaneous events at O and r register r/c later on clocks at r). O S x y z r

TRANSFORMATIONS – VECTOR ROTATIONS

• At t=0, imagine rotating 3-vector r into r’:
• This can be written

r’ O S x y z r A(P)=

cos & + (!

"(1 − cos &)

(!(# 1 − cos & − ($ sin & (!($ 1 − cos & + (# sin & (#(! 1 − cos & + ($ sin & cos & + (#

"(1 − cos &)

(#($ 1 − cos & − (! sin & ($(! 1 − cos & − (# sin & ($(# 1 − cos & + (! sin & cos & + ($

"(1 − cos &)

r’=Ar Q′- = S

./0 1

=-.Q

.

Length invariant under rotation: =20 = = −P = =⊺ |=⃗ Q|4 = =⃗ Q ⊺ =⃗ Q = ⃗ Q⊺=⊺= ⃗ Q = ⃗ Q⊺=20= ⃗ Q = ⃗ Q⊺ ⃗ Q = |⃗ Q|4 P

LORENTZ TRANSFORMATIONS AND INERTIAL FRAMES

• Inertial frames move relative to each other at constant speed, without acceleration
• In relativity, space and time coordinates of an event observed in different inertial frames are related by

Lorentz transformations t->t’(t,x,y,z), x->x’(t,x,y,z), y-> y’(t,x,y,z), z->z’(t,x,y,z)

• Frame S’ is said to be “boosted” with velocity (v,0,0) with respect to S, and has Origin O’ at (t’,vt’,0,0).

r O S x, x’ y z O’ r’ z’ y’ v S’ Vector r’ in S’ is related to r in S via Lorentz transformation

Λ5

6 =

V −VWX! −VWX! 1 + (V − 1)X!

4

−VWX" −VWX# (V − 1)X!X" (V − 1)X!X# −VWX" (V − 1)X"X! −VWX# (V − 1)X#X! 1 + (V − 1)X"

4

(V − 1)X"X# (V − 1)X#X" 1 + (V − 1)X#

4

LORENTZ TRANSFORMATIONS

• Space and time components are mixed by Lorentz boost Lµn in the direction Y

X =(nx,ny,nz) to another inertial (non-accelerating) frame, where

• The Lorentz transformation of 4-vector x/ into (x’)/ takes the form

µ = 0,1,2,3; n = 0,1,2,3 W =

7 8 ; V = 02!"

#"

'5 9 = Λ 6

5 '6

Y X = 1 Λ 6

5 =

V −VWX! −VWX! 1 + (V − 1)X!

4

−VWX" −VWX# (V − 1)X!X" (V − 1)X!X# −VWX" (V − 1)X"X! −VWX# (V − 1)X#X! 1 + (V − 1)X"

4

(V − 1)X"X# (V − 1)X#X" 1 + (V − 1)X#

4

In Einstein summation notation, an index repeated upstairs/downstairs (or vice versa) indicates a sum

• ver that index

FOUR VECTORS

0% = 1 2 , 4

Spacelike Part Timelike Part

5% = 6 2 , ⃗ 8 9% = 2:, ⃗ 9 ;% = <9% <= = > <9% <: = >2, > ⃗ ? @% = 2A, ⃗ B

LORENTZ TRANSFORMATIONS - EXAMPLE

• For a boost of velocity v = (v,0,0) of Frame S’ relative to Frame S , what is the Lorentz transformation

tensor, and how does the 4-displacement ('$, '0, '4, '1) transform? Λ6

5 =

V −VW −VW V 1 1 ('$)′ = V'$ − VW'0 ('0)′ = −VW'$ + V'0 ('4)′ = '4 ('1)′ = '1 '5 ′ = Λ 6

5 '6

THE TIME MACHINE- SPACE VS. TIME

• What are fundamental differences between space and time dimensions? Consider p. 6:
• Unlike other spacetime dimensions, time has a fixed direction (arrow of time) in which events progress,

as seen in thermodynamics

WORLDLINES, WORLDSHEETS, WORLDVOLUMES

• The worldline of a particle is the path generated by its trajectory in space and time (times the speed of

light x0 = ct)

• The worldsheet is the generalization of a worldline to trajectories of 1-D objects, developed by Leonard

Susskind to describe open and closed strings. Worldvolumes are higher dimensional generalizations. O S x y,z ct The worldlines of one particle at rest in S, and another accelerating in the positive, then negative x-direction

TIME DILATION

• Compare a light clock that undergoes a tick (round trip of a photon) in Frames S and S’
• Special relativity postulates the speed of light is the same in both stationary and moving clock
• During a tick Δ29 of the moving clock, the path length of light’s worldline is longer in S’ than in S

S x y z z’ y’ v S’ x’ ⟹ Δ2′ = VΔ2, V = 1 1 − v4/E4 h

Δ9 =

$% &

O O’

Δ9' =

$ %!((C*+"/$)! &

PROPER TIME ON A WORLDLINE

• The time measured between events in a frame in which clocks are stationary is the proper time ], and

time dilation reads

• On Worldline AB connecting Event A to Event B, the proper time is

]:+ = ^

: +

1 − v4(29)/E4<2′ <2′ = V<]

THE TIME MACHINE - PROPER TIME AND 4-VELOCITY

• Classically, the rate at which an object travels through space is dx/dt, but there is no natural way to express

a rate of passage through time.

• Consider p. 15
• How can we use relativity to describe the rate of passage through space AND time?
• In relativity, we can write the rate of travel through space AND through time as dxi/d]= V_ and dx0/d]=VE

<9% = <9&, < ⃗ 9 = 2<:, < ⃗ 9 = >2<=, < ⃗ 9 ;% = <9% <=

LENGTH CONTRACTION

• Now consider Events A, B and C for light bouncing parallel to the motion of a light clock in Frames S’

O’ S’ x’ x’ z’ O’ z’ y’ v S’ 2′:+ =? 2′+; =? y' x’ O’ v v z’ y’

A B C

L’ S’

LENGTH CONTRACTION

• Now consider Events A, B and C for light bouncing parallel to the motion of a light clock in Frames S’

O’ S’ x’ x’ z’ O’ z’ y’ v S’ 2′:+ = a9 + v2′:+ E ⟹ 2′:+ = a′ E − v 2′+; = a9 − v2′+; E ⟹ 2′+; = a′ E + v Δ29 = 2′:+ + 2′+; =

4<$ * 02%"

#"

=

4="<$ *

Δ29 = VΔ2 = V 4<

*

y' x’ O’ v v z’ y’

A B C

L’ S’ ⟹ a9 =

< =

POP QUIZ - RELATIVITY OF SIMULTANEITY

• Imagine entering a reference frame comoving with a spaceship

POP QUIZ - RELATIVITY OF SIMULTANEITY

• If two equidistant light sources turn on at the same time in the spaceship comoving frame, do you first

see light from a) Source A b) Source B c) Both Sources A and B L/2 L/2 A B

POP QUIZ - RELATIVITY OF SIMULTANEITY

• Now enter a reference frame in which the spaceship is moving at velocity v. Does the person in the

rocket first see a) Source A b) Source B c) Both Sources A and B A B v

SPACETIME INTERVAL

• We have seen in special relativity that distances, time intervals and even simultaneity depend on the

inertial reference frame in which they are measured.

• Is anything invariant under Lorentz transformation?
• The spacetime interval

is Lorentz invariant ∆5′ 2 = −(∆'$′)2 +(∆'0 ′)2 +(∆'4 ′)2 +(∆'1 ′)2 = ∆5 2

INTERSTELLAR - (39:10-40:50)

• Interstellar Min (39:10-40:50) - Cooper explains Murph will think his watch slow when he is traveling

close to the speed of light (or, as we’ll seem later, when he is near a black hole) in his mission. To what relativistic phenomenon does this pertain?

INTERSTELLAR - TIME DILATION

• Interstellar Min (39:10-40:50) - Cooper explains Murph will think his watch slow when he is traveling

close to the speed of light or is near a black hole in his mission due to time dilation.

BOHR CORRESPONDENCE PRINCIPLE

• A revised theory must agree with the previously established theory in the classical limit

RELATIVISTIC CORRECTIONS TO CLASSICAL PHYSICS – VELOCITY ADDITION

• In relativity, no object or information can travel faster than light, a fact reflected in the

relativistic law of composition of velocities

• vAB is the (1D) velocity of A with respect to B
• vBC is the (1D) velocity of B with respect to C

vBC B A C vAB v:; = v&'>v'(

0>v&'v'(

#!

POP QUIZ - RELATIVISTIC CORRECTIONS TO CLASSICAL PHYSICS – VELOCITY ADDITION

• Compare vAC to c
• What should vAC reduce to in the non-relativistic limit (vAB , vBC <<c) in terms of vAB and vBC ?

vBC=0.5c B A C vAB=0.9c v:; = v&'>v'(

0>v&'v'(

#!

POP QUIZ - RELATIVISTIC CORRECTIONS TO CLASSICAL PHYSICS – VELOCITY ADDITION

• Compare vAC to c
• What should vAC reduce to in the non-relativistic limit (vAB , vBC <<c) in terms of vAB and vBC ?

vBC=0.5c B A C vAB=0.9c v:; = v&'>v'(

0>v&'v'(

#!

v"# =

0.9 + 0.5 1 − 0.9 6 0.5 7 = 0.96557 < 7

v'( → v"$ + v$#

REVIEW OF SPECIAL RELATIVITY

• Length contraction
• Exercise: A circle of radius r=1 in its rest frame is boosted with !=2. What is its area now?
• Relativity of simultaneity
• Whether two events are simultaneous depends on the state of motion of the observer

A=cab A=c A=c/2 0.5 1 1

GENERAL THEORY OF RELATIVITY

• Einstein sought to extend his 1905 special theory of relativity to include non-inertial (accelerating)

frames

• In strong gravitational fields, such as near black holes or neutron stars, classical+special relativistic

predictions for lengths, times, frequencies, energy, etc. fail

• In a 1915 lecture at the Prussian Academy of Sciences in Berlin, Einstein unified gravity with special

relativity in the general theory of relativity

EQUIVALENCE PRINCIPLE

• Consider two labs: GraviLab and AccLab
• Is there any experiment in either lab that can distinguish the black hole from the piston?

GraviLab AccLab

EQUIVALENCE PRINCIPLE - TIDAL FORCES

• Consider two labs: GraviLab and AccLab
• Tidal forces would affect only the person in GraviLab:

GraviLab AccLab d ≈ fg Q1

EQUIVALENCE PRINCIPLE

• Consider two labs: GraviLab and AccLab
• For sufficiently small labs, there is no experiment to distinguish them

GraviLab AccLab

EQUIVALENCE PRINCIPLE

• Equivalence principle:
• Mathematically, near any point h of globally curved spacetime there exists a local coordinate

transformation of the metric tensor g56 → g?@ transforming it to the flat space metric tensor Local propermes of curved spacemme are indismnguishable from flat spacemme g?@ h = k?@

INTERSTELLAR (1:08:25-1:10:40) -TIDAL FORCES

• Interstellar Min 1:09:18 - Miller (Water) Planet 130% Earth gravity
• Interstellar Min 1:10:30 - Miller (Water) Planet mountainous waves
• .

BLACK HOLES - HISTORY

• In 1783 John Mitchell theorized that if a star’s radius is >500RSun, light could not escape it
• In 1916, Karl Schwarszchild solved the Einstein field equations for the geometry outside a spherically

symmetric, non-rotating mass

• In 1963, Roy Kerr solved the Einstein field equations for the geometry outside a spherically symmetric,

rotating mass

• In the 1970’s, black hole temperature was theorized by Hawking, among others

STRONG GRAVITY AND BLACK HOLES

• We have seen that Einstein’s special relativity postulates can warp space and time
• Can gravity warp spacetime as well?
• Yes! Dense objects change the spacetime distance formula from flat space
• STRONG gravity as one approaches singularity of infinite spacetime curvature

Singularity

BLACK HOLES - THEORY

• Objects launched from massive bodies will fall back unless their kinetic energy is at least the magnitude of

the potential energy binding them to the object:

• Note: The object’s mass m does not enter the escape speed
• Exercise: Apply the escape speed condition to a massless photon traveling at the speed of light c to find the

Schwarzschild radius Q

A , i.e., the radius in which M would have to be contained so that light cannot escape. B7)*+" 4

=

CDB E

M R ⇒ vFG8 = 2mf n m v

BLACK HOLES - THEORY

• Objects launched from massive bodies will fall back unless their kinetic energy is at least the magnitude of

the potential energy binding them to the object:

• Note: The object’s mass m does not enter the escape speed
• Exercise: Apply the escape speed condition to a massless photon traveling at the speed of light c to find the

Schwarzschild radius Q

A , i.e., the radius in which M would have to be contained so that light cannot escape.

⇒ Q

A ≡ n = 4CD *" B7)*+" 4

=

CDB E

M rS c ⇒ vFG8 = 2mf n E = 2mf n

INTERSTELLAR (1:08:25-1:10:40 & 1:46:15-1:47:28) - TIDAL FORCES

• Interstellar Min 1:11:00 - Miller (Water) Planet 130% Earth gravity
• Interstellar Min 1:11:45 - Miller (Water) Planet mountainous waves
• Interstellar 1:46:40 Romiley explains Gargantua is a ”gentle” enough singularity that a probe crossing

the horizon could survive

GRAVITATIONAL TIME DILATION AND REDSHIFT

• Gravitational time dilation
• Gravitational fields increase the oscillation period of an escaping photon and slow clocks according to
• Exercise: A visible photon leaving the vicinity of a black hole appears to a.) get redder, b.) get bluer c.) stay the

same

t$ = tH 1 − vFG84 E2 = tH 1 − 2mf QE2 = tH 1 − QE Q += = ℎr = ℎ d = ℎE s As the photon loses energy, s increases, resulting in gravitational redshift

GENERAL RELATIVITY SUMMARY

• General relativity
• Gravity warps spacetime:
• Equivalence principle: Local properties of curved spacetime are indistinguishable from flat spacetime (modulo

tidal forces)

• Gravitational time dilation
• Gravitational fields increase the oscillation period of an escaping photon and slow clocks according to

g?@ h = k?@ <t2= g56<'5<'6 t$ = tH 1 − 2mf QE2 = tH 1 − QE Q

BLACK HOLES IN ASTRONOMY

• Remnants of some stars ≳ 4MSun produce black holes when the star runs out of nuclear fuel providing
• utward pressure against gravitational collapse
• If a black holes is formed from a star that was in a binary system, it accretes the companion star,

producing x-ray radiation

• Supermassive (106-1010MSun) black holes reside in Active Galactic Nuclei
• Relativistic jets of radiating cosmic rays can be ejected from the poles of black holes in:
• Active galactic nuclei
• BH/X-Ray binaries
• Gamma ray bursts

APPROACHING THE HORIZON

• Sgr A*

53

Zooming in on Milky Way’s Sgr A* Region

Courtesy of European Southern Observatory (ESO)

?

BLACK HOLES - OBSERVATIONS

• The Event Horizon Telescope is a collection of radio antennae forming a network of intercontinental baselines

to form mm-images

• Exercise: Can you explain why baselines of radio telescopes are so long in view of the angular resolution limit

below?

• The ability to distinguish two sources at smaller angular separation increases with aperture diameter

∆PIJK= 1.22 s w LMFNOPNF EHT 2019

INTERSTELLAR-EINSTEIN RINGS

• Interstellar 2:13:24 – Shuttle approaches Gargantua with visible Einstein rings

HUBBLE’S LAW

• Hubble observed light from galaxies was far more likely to be redshifted than blueshifted,

as they were receding into an expanding universe with speed as a linear function of distance

• The current value of the Hubble constant (which is constant over space, not time) is

indicating that for every megaparsec one looks into the sky, objects recede an additional 70km/s g0 = 70 km/s Mpc Edwin Hubble 1889-1953 v = g0w, g(2) = ̇ } }(2) Scale factor a(t) compares lengths comoving with fabric of space to initial lengths in a dynamic universe

g4 + ~E4 }4 − E4 3 Λ = 8cm 3 ,, g = ̇ } }

• Solve the 1st Friedmann equation

for the scale factor a(t) in an expanding Λ -dominated Universe.

g = ±E Å 3

FATE OF THE UNIVERSE

̇ ⇒ } = <} <2 = g} = ±E Å 3 } ⇒ } 2 ~ÉQ' = É

±* S 1'

1st Friedmann Equation

FATE OF THE UNIVERSE

• The fate of the universe depends on the ratio Ω =

G G;<=> of its density relative to the critical density

• If the Universe has Ω < 1 , then it will eventually contract under its own gravity into a fiery collapse.
• If the Universe has Ω = 1 , then it will continue expanding indefinitely, but at an ever-slowing rate.
• If the Universe has Ω > 1 , then its expansion will accelerate into a cold, isolated future.

g = ̇ } } g0 = 70 km/s Mpc

:? = Ω?:@ABC, :A = ΩA:@ABC, :D= ΩD:@ABC, Ω = ΩE + ΩA + ΩD, Ω@ABC = 1 :@ABC,G = 9.47x10H$I JK

L%

1st Friedmann Equation g4 + ~E4 }4 − E4 3 Λ = 8cm 3 ,

REISS, SCHMIDT, PERLUTTER 2011 NOBEL PRIZE – ACCELERATING UNIVERSE

• A. Reiss (JHU), B. Schmidt (Australia National U.) and S. Perlmutter (Berkeley) won the 2011 Nobel Prize

in physics for observing the accelerating expansion of the Universe. Popular science summary given in: https://www.nobelprize.org/nobel_prizes/physics/laureates/2011/popular-physicsprize2011.pdf g = ̇ } } g0 = 70 km/s Mpc g4 + ~E4 }4 − E4 3 Λ = 8cm 3 , Brightness (mag) vs. distance (z) for Type Ia supernovae from observations by Brian Schmidt’s High-z Supernova Search Team (Riess et al. 1998) and Saul Perlmutter’s Supernova Cosmology Project (Perlmutter et al. 1999). Theoretical curves overlay the observations for cosmological models (ΩD, ΩT) =(1.0,0.0), (0.3,0.0), (0.3,0.7). The best fit is for the Λ–dominated Universe.

:? = Ω?:@ABC, :A = ΩA:@ABC, :D= ΩD:@ABC, Ω = ΩE + ΩA + ΩD, Ω@ABC = 1 :@ABC,G = 9.47x10H$I JK

L%

THANK YOU!