Gravitational lenses

α

β

DL

DLS

I

O

S

source

plane

lens p

lane

ηξ

ζ

θ

DS✓ = ⌘ + ⇣ = DS� �DLS↵✓ +DLS

DS↵ = �

⇠ = DL✓

Lens equation

Schwarzschild lens

↵(⇠) = �4GM

c2⇠

⇠2

✓ � ✓2E✓

= �

✓2E :=4GM

c2DLS

DSDL

↵(t) = �2G

c

2

Z⇢(t,x0)

b

b

2

✓s · ks

� se · kse

◆d

3x

0

= �4G

c

2

Z⇢(t,x0)

b

b

2d

3x

0

= �4G

c

2

Z⌃(t, ⇠0)

⇠ � ⇠

0

|⇠ � ⇠

0|2 d2⇠

0

Gravitational lenses

✓± =1

2

✓� ±

q�2 + 4✓2E

◆

✓E ⇠ 2 as M ⇠ 1012M�

⇠ 0.5mas M ⇠ M�

� ⌧ ✓E, ✓± = ±✓E + �/2

� = 0, ✓± = ±✓E Einstein ring

� � ✓E, ✓± =

(� + ✓2E/�

�✓2E/�

Gravitational lenses: Magnification

✓± =1

2

✓� ±

q�2 + 4✓2E

◆

Brightness proportional to ∫βdβdϕ at the source, ∫θdθdϕ at the observer

µ± =✓±d✓±�d�

= ±1

4

✓�p

�2 + 4✓2E+

p�2 + 4✓2E

�± 2

◆.

|µ+|+ |µ�| =1

2

✓�p

�2 + 4✓2E+

p�2 + 4✓2E

�

◆> 1

Microlensing

Gravitational lenses: Distortion

β

2χ

b

a=

�p�2 + 4✓2E

(� ⌧ �)

Φ

�c = 2⇥Etan(�/2)

1� sin(�/2)

Gravitational lenses

Post-Newtonian time

d⌧ =

1� 1

c2

✓1

2v̄2 + U

◆+O(c�4)

�dt̄

=

1� 1

c2

✓1

2v2 + v · (! ⇥ r) + �

◆+O(c�4)

�dt̄

� = U +1

2!

2(x2 + y

2)

A

B

For travelling clock with v << c

⌧A =

✓1� �

geoid

c2

◆t̄

⌧B =

✓1� �

geoid

c2

◆t̄+ ⌧0B

Geoid potential

Sagnac effect: 207 ns for closed path at the equator

Clock synchronization on the Earth

Two clocks at rest

⌧trav

= ⌧A(t̄1) +

✓1� �

geoid

c2

◆(t̄

2

� t̄1

)� 1

c2

Z B

A(! ⇥ r) · dr

⌧0B = � 1

c2

Z B

A(! ⇥ r) · dr

d⌧ground

=

✓1� �

geoid

c2

◆dt̄

d⌧sat =

1� 1

c2

✓1

2v2 + U

◆�dt̄

=

✓1� 3

2

Gm

c2a

◆dt̄

d⌧sat

✓1 +

�geoid

c2� 3

2

Gm

c2a

◆d⌧

ground

Post-Newtonian time: GPS

⇠ Gm

c2a�

⇠ 6.9⇥ 10�10

⇠ 3

2

Gm

c2(4.2a�)

⇠ 2.5⇥ 10�10

for GPS satellites

General Relativity and Daily Life The Global Positioning System (GPS)

Navigation Requirement: 15 m ➭ 50ns

Relativistic effects: 39,000 ns per day!

GR = 46,000 SR = -7,000

Relativity mustbe taken into account, for GPSto function