1. Harrison B. Prosper Florida State University YSP
Relativity 2
Harrison B. Prosper
Florida State University
YSP

2. Topics Part 1 Recap Mapping Spacetime When Is Now? Part 2
Distances in Spacetime
Paradoxes
Summary

3. Recap Einstein’s theory is based on two postulates:
Principle of relativity: The laws of physics are the same in all inertial (that is, non-accelerating) frames of reference.
Constancy of the speed of light: The speed of light in vacuum is independent of the motion of the light source.

4. Spacetime Diagrams Events can be represented as points in a
Events with the
same time values,
such as events A and
B, are said to be
simultaneous
Time A B
Space

5. Event: A place at a given time Spacetime: The set of all events
Earth’s Time Axis
3000 AD D C
now B
(t,x,y,z)
2500 AD A
2000 AD y x O

6. Mapping spacetime

7. Mapping Spacetime – I tC tB tD tA tB = γ tD tD = κ tA C
light
Starship’s worldline
v = speed
= BD / OB
= x / tB
Earth’s
worldline B tB
c = BD / AB D A tA
light tD
Line of simultaneity
“Now”
tB = γ tD
tD = κ tA O

8. Mapping Spacetime - II tC tB tD tA κ-factor Relates elapsed times C
from a common event O
to two events A and D
that can be connected
by a light ray.
γ-factor
to two events B and D
that are judged
simultaneous in one
of the frames. A B C tA tD D O tC tB

9. κ and γ Factors
Relativistic Doppler Factor
Dilation Factor

10. κ and γ Factors Relativistic Doppler Factor
Problem 1: derive the formula for κ.
Problem 2: light of 500 nm wavelength is emitted by a starship, but received on Earth at a wavelength of 600 nm. What is the relative speed between the Earth and the starship?

11. 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

12. When Is Now? - II “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

13. Worlds in Collision
T. J. Cox and Abraham Loeb

14. Distances in spacetime

15. The Metric dl dy dx The distance between points O and P is given by:
OB2 + BP2 = OP2 = OA2 + AP2
OP2 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

16. The Metric z AC = r dφ r CB = dr AB = dl θ φ y C x B AΔφ C B A
The metric in spherical polar coordinates
(r, θ, φ)
Consider the spatial plane θ = 90o
AC = r dφ
CB = dr
AB = dl

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

18. 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

19. 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

20. Paradoxes

21. The Pole and the Barn Paradox
A 20 m pole is carried so fast that it contracts to 10 m in the frame of reference of a 10 m long barn with an open front door. Consequently, the pole can fit within the barn for an instant, whereupon the back door is swung open. But in the pole’s frame of reference, the barn is only 5 m long, so the pole cannot possibly fit in the barn! Resolve the paradox (hint: draw a spacetime diagram)

22. Betty and Ann Δt = (0.8) (8y) = 6.4 years 8 light years (ly) β = 0.8
2053.6
Betty’s Now in 2056
Betty and Ann
Betty’s worldline
2060
Ann’s Now in 2060
Δt = (0.8) (8y)
= 6.4 years
2056
Ann’s Worldline
Ann’s Now in 2050
2050
8 light years (ly)
β = 0.8
1/γ = 0.6
Example from About Time by Paul Davies

23. Twin Paradox Betty 8 years younger! Betty’s Now in 2062
8 light years
2050
2060
2070
Ann’s Now in 2060
Ann’s Now in 2050
Ann’s Now in 2070
2068
sent by Betty
in 2056, Betty’s time,
received by Ann in
2068, Ann’s time.
2053.6
Betty’s Now in 2050
Betty’s Now in 2056
2052
sent by Ann
in 2052, Ann’s time,
received by Betty in
2056, Betty’s time.

24. Temporal Paradox 8 light years 8 light years A Betty’s Now in 2050
2060
Ann’s Now in 2060
Super-luminal signal sent
to star A in 2050, arriving
in 2056 according to Betty.
But for Ann, signal sent in
2050 arrives in 2047!
Ann’s Worldline
2053.6
8 light years
Ann and Betty’s Now in
2047.2 A
8 light years
Ann’s Now in 2050
2050
2048
Super-luminal signal sent from A
arrives in 2048, preventing signal
sent in 2050!

25. How To Make A Time Machine!

26. The Two Spaceship Problem
During the acceleration, do the captains measure a fixed, or changing, distance between the spaceships? Explain!
Earth’s worldline
Captain
Vivian
Captain
Luke t2 t1 x

27. Summary
There is no absolute “now”. Each of us has our own “now” determined by how we move about
Super-luminal travel, within a simply-connected spacetime, would lead to temporal paradoxes
The time between events depends on the path taken through spacetime, with an inertial (non-accelerating) path yielding the longest time

28. Appendix

29. Proper Distance By definition: The proper distance is the
spatial separation between
two simultaneous events.
C & D are simultaneous in the Red
frame of reference.
E & D are simultaneous in the Yellow C D x’ E x A B

30. The Lorentz Transformationx P O
Define
Δt = tBC + tCP
Δx = xOA + xAB t' x' C
What is the
interval
between
event O
and
event P?
Δx' = xOQ
Δt' = tQP B A Q
tCP = tQP /γ
tBC = β Δx / c
xOA = xOQ / γ
xAB = v Δt

31. The Lorentz Transformationx' C t x P O B A Q
tBC = v Δx/ c2
tCP = tQP / γ
xOA = xOQ / γ
xAB = v Δt
We obtain the Lorentz transformation
(Δx, Δt) → (Δx', Δt')
Δx' = γ (Δx – β cΔt)
cΔt' = γ (cΔt – β Δx)
The interval from O to P is
OP2 = (cΔt)2 – (Δx)2
= (cΔt')2 – (Δx’)2

32. Problem 5
Compute the spacetime distance (ds) between the following events: 1. event 1: solar flare on Sun (in Earth’s) now. event 2: a rainstorm here, 7 (Earth) minutes later. (Give answer in light-minutes.) 2. event 1: the fall of Alexandria in 640 AD event 2: Tycho’s supernova seen in 1572 AD (the star was then 7,500 ly from Earth). (Give answer in light-years.)