Prelude: GR for the Common Man

Intro Cosmology Short Course

Lecture 1

Paul Stankus, ORNL

What is Calvin & Hobbes’ primary misconception?

Subjective time -- “proper time” -- is a fundamental physical

observable, independent of coordinates

Same path through space-time

Same subjective elapsed time

xt

(dt, dx)

€

dτProper

Time

{2 = dx μ gμν x 0, x1,x 2, x 3

( ) dxν

μ ,ν = 0

3

∑

B. Riemann German

Formalized non-Euclidean geometry (1854)

g

Metric Tensor

€

(dt,dx,dy,dz) at some point (t, x, y,z)

€

(dx 0,dx1,dx 2,dx 3) at (x 0, x1, x 2, x 3)

Newtonian:

Most general:

x

t

Bank News Stand

YouMe

Car

€

Δt ≈ Δτfor all

paths

Assuming Newton found parking….

Galilean/ Newtonian

€

dτ 2 ≈ dt 2 g00 ≈1 g0i ,gij ≈ 0

x

t

UpDownBasement detector

Down-going muon decays

Stopped muon decays

2.2 sec

Cosmic ray lab

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.Lab

Basement

Down-going photon

1. Down-going muons v < c

2. Lifetime of down-going muons > 2.2 secWe

observe:

x

tCurve of constant Δ=2.2 sec

H. Minkowski German

“Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality" (1907)€

dτ 2 = dt 2 − dx 2 /c 2

x’

t’Rest frame of lab-stopped muon

Rest frame of down-going muon

€

dτ 2 = dt '( )2

− dx'( )2

/c 2

€

gμν =

1 0 0 0

0 −1/c 2 0 0

0 0 −1/c 2 0

0 0 0 −1/c 2

⎡

⎣

⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥

in any and all intertial frames

The Twin “Paradox” made easy

x

t

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are needed to see this picture.

Castor inbound

T

X

Casto

r out

boun

d

Pollux at rest

€

Pollux = 2 ΔT 2 = 2ΔT

τ Castor = 2 ΔT 2 − ΔX 2 c 2 < τ Pollux

It’s just that simple!

New, generalized laws of motion

1. In getting from A to B, all free-falling objects will follow the path of maximal proper time (“geodesic”).

2. Photons follow “null” paths of zero proper time.

x

t

A

B

€

dτ 2 = 0

dt 2 − dx 2 c 2 = 0

dx

dt= ±c

Recovering Newton’s First Law

x

t

A

B

x

t

A

B

T

T

X X

€

ΔAB (ε) = T 2 − (X + ε)2 c 2 + T 2 − (X −ε)2 c 2

Maximized with ε = 0, ie a straight line

⇒ Velocity remains constant

An object at rest tending to

remain at rest

Gravitational Red Shift

x

t

UpDown

Source Downstairs

Detector Upstairs

Photon period

tDownstairs

Downstairs

Photon period tUpstairs

Upstairs

€

ΔtDownstairs = ΔtUpstairs

but

Δτ Downstairs < Δτ Upstairs

so dτ

dt ≠ constant over x

Let dτ

dt=1+ φ(x) c 2

€

Resulting metric :

dτ 2 = 1+ φ(x) c 2[ ]

2dt 2 − dx 2 c 2

Non-inertial frame -- curved space!

Motion in curved 1+1D space

x

t

A

B

h

T

T

€

dτ 2 = 1+ φ(x) c 2[ ]

2dt 2 − dx 2 c 2

€

so a =Δv

T=

(−h /T) − (h /T)

T= −φ'(h /2)€

maximize [1+ φ(h /2) c 2]2T 2 − h2 c 2

≈ [1+ 2φ(h /2) c 2]T 2 − h2 c 2

find h = (1/2) φ'(h /2)T 2

€

dτ AB h( ) = 2 [1+ φ(h /2) c 2]2T 2 − h2 c 2

Conservative (Newtonian) Potential!

Albert Einstein German

General Theory of Relativity (1915)

Isaac Newton British

Universal Theory of Gravitation (1687)

€

rF (

r x ) =

GMm

r2ˆ r

r a (

r x ) =

r F

m=

GM

r2ˆ r = −

v ∇ −

GM

r

⎛

⎝ ⎜

⎞

⎠ ⎟= −

r ∇φ(

r x )

€

dτ 2 = 1+ φ(r x ) c 2

[ ]2dt 2 − d

r x 2 c 2

+maximum proper time principal

⇒r a (

r x ) = −

v ∇φ(

r x ) for φ /c 2 <<1

Metric: a local property

Force law: action at a distance

Points to take home

• Subjective/proper time as the fundamental observable

• Central role of the metric

• Free-fall paths maximize subjective time

• Minkowski metric for empty space recovers Newton’s 1st law

• Slightly curved space reproduces Newtonian gravitation