Lecture 16 GR – Curved Spaces

ASTR 340 Fall 2006 Dennis Papadopoulos

GR Framework

GR EFFECTS IN THE SOLAR SYSTEM

•

Orbit of Mercury:

Mercury does not move in perfect ellipse but precesses-> Vulcan?

sun

Mercury Precession

• Effect called “precession of perihelion”.

• Effect small - orbit twists by 5600 arc-seconds (1.56 degrees) per century – With Newtonian gravity, can explain 5557 arc-seconds/century as due to • Gravitational effect of other planets, • deformation of the Sun, • non-inertial nature of Earth’s frame – But still leaves 43 arc-seconds per century unexplained… • Using GR, Einstein predicted (with no fiddling!) that Mercury should precess 43 arcseconds per century!

Gravitational Lensing

Gravitational Lensing

Also have light bending by distant galaxy clusters: “giant lenses” in the sky

Gravitational micro-lensing

• Individual stars can also make a gravitational lens…

microlensing

.

• Suppose we… – Look at a distant star in our galaxy – Another massive (but dark) star passes in front… From web site of Ned Wright (UCLA) – Causes apparent increases in brightness of stellar image

Gravitational time dilation has practical importance!

• Global Positioning System (GPS) – System of satellites that emit timing signals – Detector on Earth receives signals – Can figure out position on Earth’s surface by measuring time delay between signals from different satellite (light travel time gives distance to satellite) – Need to measure time of signal from satellite very well!

– Satellites are at varying heights; clocks run at varying rates • If GR effects were not included, computed GPS positions would drift from true position by

kilometers per day

!

Escape Speed

0 R E   1 2 2 ( ) 

G M m E r



const

.

r 1

PE

  

GM m E r

1 2

for

 (

E

) 

G M m E R E

 1 2 0

v E

 2

G M E R E

 2

gR E

(   )

m eff E

 

hf E

/

c

2

hf A

 (

hf A

/

c

2 )

gH



hf B f B



f A

( 1 

gH

/

c

2 )

Fig. 3-25, p. 95

hf



GM R s hf

(

c

2 ) 

h f



f

 

f

( 1 

GM R s c

2 )

Fig. 3-26, p. 96

Gravitational Redshift

z

   0  2  1 2 (

v E

2

c

2  1 )

E f

 

f

(1 

GM R c s

2 ) 

f

[1  1 2 (

v E c

2 ) ] There is a ½ factor in error because we used classical arguments

Einstein’s tower • So far, we have ignored the effects of gravity on light. Is this really okay?? • Consider another thought experiment, to test whether

light

can be

unaffected by gravity.

• Consider a tower on Earth – Shine a light ray from bottom to top – When light gets to top, turn its energy into mass.

– Then drop mass to bottom of tower, in Earth’s gravity field – Then turn it back into energy

• If we could do this, then we could get energy from nothing!

– Original energy in light beam = E start – Thus, mass created at top is m=E/c 2 – Then drop mass… at bottom of tower it has picked up speed (and energy) due to the effects of gravitational field.

– When we turn it back into energy, we have E end =E start +E grav – But, we started off with only E start energy! We’re rich!

– we have made

Remember the tower…

• Light beam must lose energy as it climbs up – So…frequency must decrease – i.e., light is redshifted.

– Gravitational redshifting • Imagine a clock based on frequency of laser light… – 1 “tick” = time taken for fixed number of crests to pass – Gravitational redshifting slows down the clock.

–

Clocks in gravitational fields must run slowly

t grav

  1

GM c

2

r

 

t space

if gravitatio nal field is " weak"

Tidal Effects

Differences between accelerating and gravitational frames – Non locality

Time dilation in GR

v

 

r

ACCELERATION AND TIME

Slim and Jim compare their watches while Jim crawls slowly along the radius. Slim’s clock runs slower since he was always moving faster than Jim. Example of warped time, rate of passage differs from location to location

ACCELERATION AND WARPING OF SPACE/TIME

Measure radius and circumference with no spin you find their ratio equal circumf/radius=

2p6.28.

Do it again when the wheel is spinning.

Radius the same but circumference longer Ratio> 6.28

CURVED SPACE-TIME

• Einstein pondered several things… – Success of Special Relativity showed that space & time were closely interlinked – The “tower thought experiment” suggested that free-fall observers are (locally) free of effects of gravity: frequency of light they observe does not change as they accelerate – He wanted to say that gravity was an illusion caused by the fact that we live in an accelerating frame… – … but there is no

single

accelerating frame that works! Somehow, you need to stick together frames of reference that are accelerating in different directions

• Einstein’s proposal – 4-dimensional space-time is “curved,” not flat • Example: surface of sphere is curved 2D space; surface of football field is flat 2D space – Free-falling objects move on “geodesics” through curved space-time (generalizations of straight lines in flat space).

– The curvature (bending) of space-time is produced by matter and energy • What is a geodesic?

– Shortest path between two points on a surface – E.G. path flown by an aircraft between cities on the globe – Geodesics that start parallel can converge or diverge (or even cross).

On Globe…

• Constant-longitude lines (meridians) are geodesics • Constant-latitude lines (parallels) are

not

Geometry of space

Geodesics on sphere and torus

Hyperbolic space

• Two-dimensional version of a hyperbola - a “saddle” • Geodesics diverge

How does matter “warp” space?

• Use two-dimensional space as an analogy: think of how rubber sheet is affected by weights • Any weight causes sheet to sag locally • Amount that sheet sags depends on how heavy weight is From web site of UCSD

Effect of matter on coordinates

• Lines that would be straight become curved (

to external observer

) when sheet is “weighted”

How are orbits affected?

• Marble would follow straight line if weight were not there • Marble’s orbit becomes curved path because weight warps space

Applied Mathematics Dept, Southampton University

Warping of space by Sun’s gravity

• Light rays follow geodesics in warped space

THE GENERAL THEORY OF RELATIVITY

• Within a free-falling frame, the Special Theory of Relativity applies.

• Free-falling particles/observers move on geodesics through curved space-time • The distribution of matter and energy determines how space-time is curved.

“Space-time curvature tells matter/energy how to move.

Matter/energy tells space-time how to curve.”

G

 8 p

G

T

c

4 • Notes: – The Einstein curvature tensor “

G

” is mathematical object describing curvature of 4-D space-time.

– The Stress-Energy tensor “

T

” is mathematical object describing distribution of mass/energy.

– Newton’s constant of gravitation “G” and the speed of light “c” appear as fundamental constants in this equation. – This is actually a horrendous set of 10 coupled non-linear partial differential equations!!

• For weak gravitational fields, this gives Newton’s law of gravitation.