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

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