prelude: gr for the common man
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Prelude: GR for the Common Man. Intro Cosmology Short Course Lecture 1 Paul Stankus, ORNL. Same path through space-time. Same subjective elapsed time. What is Calvin & Hobbes’ primary misconception?. - PowerPoint PPT PresentationTRANSCRIPT
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