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Arvind Borde / MTH675, Unit 18: Special Relativity IV
-1. Inertial Frames
Class of frames which
a) Move uniformly wrt each other, and in which
b) Newton’s First Law holds.
(“No force, no acceleration.”)
1
0. Spacetime Understanding of Length Contraction
Rod at rest in the hatted frame is seen as con-
tracted from the unhatted frame.2
1. More on Proper Time
(1) In 2d (1 space, 1 time), if (t1, x1) = (0, 0)
(aka “the origin”) and (t2, x2) = (t, x) what are
∆t and ∆x? ∆t = t and ∆x = x .
(2) If c = 1, what is the squared proper time
interval between (0, 0) and (t, x)?
3
(3) In 2d spacetime, with c = 1, plot the null
lines through (0, 0), the line with t = 1, and the
line with that (squared) proper time equal to 1.
4
What do you recognize in Minkowski’s diagram:
5
2. Matrix Language
Because both the Galilean and Lorentz transfor-
mations are linear, they can be expressed via ma-
trices.
6
ADDITIONAL NOTES
1
1
Unit 18, Slides 7–12 Arvind Borde
The Galilean World
If a system is moving with velocity ~v = (v1, v2, v3),
the transformation from a “stationary” system istx1x2x3
=
1 0 0 0−v1 1 0 0−v2 0 1 0−v3 0 0 1
tx1x2x3
7
(4) Multiply the matrices and write this as a set
of equations.
8
The Lorentz-Minkowski World
If a hatted system is moving with respect to an-
other, the 2d (and nd) Lorentz transformation can
also be expressed in matrix form:(tx
)= γ
(1 −v/c2−v 1
)(tx
)= γ
(t− vx/c2−vt+ x
)9
3. The Metric
We’ve seen (U10) that(ds
dt
)2= g11
(du1
dt
)2+ 2g12
(du1
dt
)(du2
dt
)+ g22
(du2
dt
)2often abbreviated (“differential” notation) as
ds2 = g11du12 + 2g12du
1du2 + g22du22
10
For example,
2d sphere of radius r:
ds2 = r2dθ2 + r2 sin2 θdφ2
2d plane in Cartesian coordinates:
ds2 = dx2 + dy2
11
Arclength of a curve α(t):
s(t) =
∫ t
a
‖α′(t)‖ =
∫ t
a
√(ds
dt
)2dt
Or, symbolically,
s =
∫α
√g11du21 + 2g12du1du2 + g22du22
Knowing gij allows you to find arc lengths.
12
ADDITIONAL NOTES
2
2
MTH675 Unit 18, Slides 13–18
The metric gij allows you to determine inner prod-
ucts (“dot products”) of vectors.
If ~V and ~W are vectors with components {V i}and {W i}, then their inner product is
The norm of a vector is its inner product with
itself.
13
If you think of the components of an nd vector as
a column matrix
V i =
V 1
V 2
...V n
and of Vi ≡ gijV j as a row
(V1 V2 . . . vn )
then you can calculate “ds2” as follows:14
3d Pythagorean metric
ds2 =
1 0 00 1 00 0 1
dx1
dx2
dx3
dx1
dx2
dx3
15
(5) Multiply and verify.
16
Minkowski metric:
ds2 =1 0 0 00 −1 0 00 0 −1 00 0 0 −1
cdtdxdydz
cdtdxdydz
17
(6) Multiply and verify.
18
ADDITIONAL NOTES
3
3
Unit 18, Slides 19–24 Arvind Borde
Both the Pythagorean and Minkowski metrics above
are constant.
Metrics such as these are called flat metrics, be-
cause the curvatures that can be computed from
them are zero.
19
From U13:
Christoffel symbols (second kind):
Γmij = gmkΓijk =1
2gmk
(∂gik∂uj
+∂gkj∂ul− ∂gij
∂uk
)Reimann-Christoffel curvature tensor:
Rmikj ≡∂Γmij∂uk
− ∂Γmik∂uj
+ ΓnijΓmnk − ΓnikΓmnj
20
4. Spacetime: A Summary
Special relativity is clarified by Minkowski’s space-
time: a 4-d (or n-d) entity unifying space and time.
If spacetime slope is |∆t/∆x|, we classify
Slope > 1/c: Timelike (worldlines of massive things)
Slope < 1/c: Spacelike (worldlines of nothing[?])
Slope = 1/c: Null (worldlines of massless things)
21
The proper time is ∆s/c (∆s is proper interval)
and is the physical time measured along an ob-
server’s worldline.
∆s2 > 0: Timelike (worldlines of massive things)
∆s2 < 0: Spacelike (worldlines of nothing[?])
∆s2 = 0: Null (worldlines of massless things)
22
5. Resolving “Paradoxes”
Here’s a sketch of a par-
ticular “twin paradox sit-
uation.”
Assume, distance is in l-
y and time in y, so c = 123
(7) From the diagram on the previous slide, cal-
culate the proper times for both the stay at home
twin and the rocketing twin, from the earth pov.
Stay at home:
Rocketeer:
24
ADDITIONAL NOTES
4
4
MTH675 Unit 18, Slides 25–30
(8) Do the same from the pov of a frame attached
to the outgoing twin.
25 26
Rocket time:
27
(9) Einstein’s Train: A train passes by you with
a guard in the middle. Just as the guard passes
you, flashes of light emitted from the front and the
back of the train reach both you and the guard.
Both of you agree on this. Do you agree on when
the flashes were emitted?
28
Simultaneous arrival of the signals.
29
Let the guard’s frame be “unhatted,” and the “rest
length” of the train be 2`.
Let t = 0 = t be the instant when both you and
the guard cross.
In the guard’s frame the signals are emitted at
xf = −`, xb = `, t = −`/c
30
ADDITIONAL NOTES
5
5
Unit 18, Slides 31–36 Arvind Borde
Apply the LTs:
t = γ(t− vx/c2), x = γ(x− vt)In your frame:
tb = γ
(−`c− v`
c2
)= −γ
c
(1 +
v
c
)`
tf = γ
(−`c
+v`
c2
)= −γ
c
(1− v
c
)`
So tb < tf .
31
Organizers' POV
Your POV32
6. Beyond Special Relativity
Einstein used Minkowski’s idea of spacetime to fur-
ther develop the theory of relativity.
When developing Special Relativity, Einstein had
several parallel investigators – Lorentz, Poincare,
and others.
When developing its extension, General Relativity,
Einstein initially acted alone.33
After many false starts and finishes, Einstein hit
upon his final formulation in 1915.
He summarized his journey and where it had led
him in his landmark paper of 1916:
34
35 36
ADDITIONAL NOTES
6
6
MTH675 Unit 18, Slides 37–42
Einstein argued that it’s necessary to extend spe-
cial relativity because it’s inherently defective.
This defect was present in Galilean relativity (New-
tonian physics), as well, but had gone unresolved
and often unnoticed for hundreds of years.
37 38
S2 S2
S1 S1
S1 pov S2 pov
39
Einstein argues that Newtonian physicists would
say frame 1, in which S1 is at rest, is inertial, and
frame 2 is not, but that decision would only be
made after observing what’s happening.
40
After some further discussion of the asymmetry
in the appearance of two fluid spheres in relative
rotation, Einstein concludes
41
Einstein goes on with a further “thought experi-
ment” (the Einstein elevator):
42
ADDITIONAL NOTES
7
7
Unit 18, Slides 43–48 Arvind Borde
xaccelerated up
Unhatted POV
yaccelerated down
Hatted POV43
Uniform downward acceleration, such as what you
see in the hatted frame, is exactly what you expect,
though, in a uniform gravitational field.
44
This led Einstein to an important conclusion:
Having argued that it would be more satisfactory
that all reference frames be treated as equal, Ein-
stein further argues that such a theory must incor-
porate gravitation.
45
Einstein goes on to a second important argument,
in order to make the case that a general theory
of relativity would need to incorporate gravitation,
and non-Euclidean geometry.
He imagines a system K and a system K ′ rotating
with respect to it. He further imagines measuring
the circumferences and diameters of circles in each
system:46
47Length contraction in direction of rotation; none in radialdirection.48
ADDITIONAL NOTES
8
8
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