1. Harrison B. Prosper Florida State University YSP
Relativity 3
Harrison B. Prosper
Florida State University
YSP

2. Topics Recap – When is now? Distances in spacetime Gravity
The Global Positioning System

3. Recap – When is now?

4. When is Now? tB tD Δt = tB - tE tE Events B and D
are simultaneous for Earth so
tD = tB / γ tB tE
But events D and E
are simultaneous
for the starship so
tE = tD / γ E
Line of simultaneity B tD
Δt = tB - tE D
Line of simultaneity O

5. When is Now? “Nows” do not coincide Δt = tB - tE tBtD D O tB
Line of simultaneity
“Nows” do not coincide
Δt = tB - tE
Line of simultaneity
Writing distance between
B and D as x = BD
the temporal discrepancy
is given by
Problem 4: estimate Δt
between the Milky Way and Andromeda, assuming
a relative speed between the galaxies of 120 km/s
Problem 3: derive Δt

6. When is Now? ct ct' x' x' x α B D E worldline of a photon
emitted from O x' θ θ

7. Distances in spacetime

8. Distances in Space dl dy dxQ O B A dl dx dy
The distance between points O and Q is given by:
OB2 + BQ2 = OQ2 = OA2 + AQ2
OQ2 is said to be invariant.
The formula
dl2 = dx2 + dy2
for computing dl2 is called a metric
In 3-D, this becomes
dl2 = dx2 + dy2 + dz2

9. Distances in Spacetimect x P O
ct' x' C
What is the
“distance”
between
event O
and
event P?
(x, ct)
(x', ct') B A Q
What is BC?
What is AB?

10. Distances in SpacetimeO Q ds dl
cdt
Suppose that O and P are events.
How far apart are they in spacetime?
First guess
ds2 = (cdt)2 + dl2
However, this does not work for spacetime!
In 1908, Hermann Minkowski showed
that the correct expression is
ds2 = (cdt)2 – dl2
ds2 is called the interval
Hermann Minkowski

11. The Interval
In general, the interval ds2 between any two events is either
timelike
ds2 = (cdt)2 – dl2 cdt > dl or
spacelike
ds2 = dl2 – (cdt)2 dl > cdt
null
ds2 = (cdt)2 – dl2 = 0 dl = cdt

12. 1. Which is the longest side and which is the shortest side?x ct A B C E F D 5 3 6
2. Which path is longer, D to F or D to E to F?
units: light-seconds
from Gravity by James B. Hartle

13. Gravity

14. Gravity All objects fall with the same acceleration G. Galileo
1564 –1642

15. The Principle of Equivalence
A person falling off a building
experiences no gravity!
“The happiest thought of my life”
Albert Einstein (1907)

16. The Principle of Equivalence
free space view
inside
view
outside
view
Gravity is curved spacetime!

17. General Relativity (1915) Bending of light
Precession of Mercury’s orbit
Sir Arthur Eddington
Eclipse Expeditions 1919

18. Metric in Spherical Polar Coordinatesx φ θ z y r Δφ C B A
Parameters of spherical polar coordinates
(r, θ, φ)
Consider the spatial plane θ = 90o
AC = r dφ
CB = dr
AB = dl
The metric in spacetime is
therefore,

19. Spacetime around a Spherical Star
In 1916, Karl Schwarzschild found the first exact solution
of Einstein’s equations:
Karl
Schwarzschild φ r

20. Spatial Geometry and Proper Distance
The proper distance between any two events is their spacetime separation ds when the time difference dt between the events is zero in a
given frame of reference. B
dt = 0 A

21. Spatial Geometry and Proper Distance
Consider the proper distance along the arc of a circular trajectory, in the Schwarzschild spacetime geometry:
dr = 0
dt = 0 ds dφ
for a complete circuit. Note: the
spacetime separation = proper distance
in this case

22. Spatial Geometry and Proper Distance
Now consider a radial trajectory, dφ = 0.
Then, the separation ds is given by r
Again, in this case, the
spacetime separation = proper distance

23. Spatial Geometry and Proper Distance
Problem: Suppose we traveled along a radial trajectory from the Earth to the Sun’s photosphere. How far would we have traveled? That is, what is the proper distance? Wolfram Mathematica Online Integrator r

24. Black Holes and wormholes

25. Black Holes Consider hovering near a black hole at a fixed radius r.
How would your elapsed time be related to the elapsed time of someone far away?
near
far away

26. Visual Distortions near a Black Hole

27. Wormholes The Morris-Thorne metric (θ = 90o) a = Throat radius
Problem 5:
Assuming a = 1 km, how far must you
walk to get to the center of the
wormhole? Assume you start at an “r”-distance
that corresponds to a circumference of
2π×10 km. How does the proper distance
compare with the “r-distance”?

28. The global positioning system

29. The Global Positioning System
What is it ?
A system of 24 satellites in orbit about Earth that provides accurate world-wide navigation
Each satellite contains an atomic clock, accurate to ~ 1 nanosecond
Each satellite emits a unique signal giving its position

30. GPS – Orbits Period: 12 hours Orbital radius: 26,600 km
Six orbital planes 60o apart

31. GPS – Principle 1 2 3
ct1
ct2
ct3
You are here!

32. GPS – Circular Orbits
For circular orbits, r does not change, so dr = 0. Therefore,
Now divide by (cdt)2, and noting that v = r dφ/dt, the tangential speed measured by an observer far away, then the elapsed time dτ at any given radius r is given by

33. vsatellite vEarth rsatellite GPS Clocks rEarth t = time far from Earth
τ = time at given radius r
rS = Schwarzschild radius
rEarth
Problem 6:
How fast (or slow) does the
satellite clock run per day
relative to the Earth clocks?
Give answer in nanoseconds

34. Summary The interval between events is invariant.
A timelike interval measures the elapsed time along a worldline.
Gravity is warped spacetime
Time is slowed by gravity