Physics 73 notes

U P K E M

M E M B E R S H I P A C A D E M I C D E V E L O P M E N T

Physics 732nd Long Exam Reviewer

Heavily based on Arciaga Notes

Preface

This handout is intended as a reviewer only andshould not be substituted for a complete lecture,or used as a reference material. The goal of thisreviewer is to refresh the student on the conceptsand techniques in one reading. But this is morethan enough to replace your notes :)

1 Principle of Relativity

1.1 Reference Frames

Definition (Reference frame). A reference frameis simply a coordinate system attached to a par-ticular observer.

Definition (Inertial frame of reference). An in-ertial frame of reference is a reference frame inwhich Newtons 1st law is valid

An object at rest, remains at rest. An objectin motion, stays in motion.

No change in speed or direction (non-acceleratingframe).

No gravitational force is felt (Earth cannotbe a true IRF, but is approximated as such)

1.2 Newtonian Relativity

Laws of mechanics are preserved in all IRFs.

This is familiar to you already, recall relativevelocities from Physics 71.

Invalid for speeds near the speed of light.

1.3 Special Relativity

1.3.1 First Postulate

All laws (their forms, not values of parameters)of Physics are the same in every inertial referenceframe.

1.3.2 Second Postulate

The speed of light in a vacuum is constant inall inertial frame of reference and is independentof the motion of the source.

Examples. Fixed quantities

Numerical value of the speed of light in avacuum

Value of the charge on the electron

Order of the elements in the Periodic Table

Newtons First Law of Motion

Examples. Examples of variable quantities

Speed

Time between two events

1

University of the Philippines Chemical Engineering Society, Inc. (UP KEM)Physics 73 - 2nd Long Exam

Kinetic energy

Force

Electric field

Magnetic field

1.4 Natural Units

To elegantly simplify particular algebraic expres-sions appearing in the laws of physics (particularlyrelativity here), we deal with natural units.

Same unit for space and time

Conversion factor : c = 299 798 482m/s

In natural units, c = 1

vnat =Xnattnat

= 1c

X

t

tnat = ct

Speed is unitless

Example. Convert 6.21 107m/s to natural units

6.21 1017m/s3 108m/s = 0.207

Example. Convert 20min to meters

20min(

60 s1min

)(3 108 ms

)= 3.6 1011m

Remark. From here on, all quantities will bein natural units.

2 Events and Measurements

Definition (Event). An event is an object thathas spacetime coordinates (t, x).

Remark. In three spatial dimensions, it is (t, x, y, z)

2.1 Invariance of the Interval

Definition (Spacetime Interval). The interval be-tween two events happening at some points (ta, xa)and (tb, xb). The interval, s, is defined by

s2 = t2 x2 = (t)2 (x)2 (1)

x = xb xa

t = tb ta

With the denoting an observation made in an-other frame of reference. Spacetime interval is in-variant, i.e. observations in two different IRFshas the same spacetime interval.

Table 2.1: Comparison of two spaces

Flat spacetime vs Euclidean space

Events Places

(t, x, y, z) (x, y, z)

invariant interval invariant distance(s2 = c2t2 x2) (d2 = x2 + y2)

2.2 Spacetime Diagrams

A spacetime diagram is a helpful visual represen-tation of events in spacetime. Note the following:

In one spatial dimension, the vertical axis ist while the horizontal is x.

An event is a point, with coordinates (t, x).

A worldline is a curve on the diagram. Thisrepresents the history of events (refer to fig-ure 2.1).

Classifications of spacetime intervals:

Timelike : t2 x2 > 0 Spacellike : t2 x2 < 0 Light-like : t2 x2 = 0

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event

worldline

x

t

Figure 2.1: An example of a spacetime diagram

No particle can travel at and exceed the speedof light light. That is:v

c

< 1Denote this ratio . It is related to the in-verse of the slope of the worldline in a space-time diagram. (or the angle between the tand t axis is tan1 )

Particles should have timelike worldlines.

The locus of all event points for a given in-terval is a hyperbola (figure 2.2).

k

x

t

Figure 2.2: Locus of events for an invariant interval = k2

One unit of time in the t and t axes areconnected by a horizontal parabola.

One unit of position in the x and x axes areconnected by a vertical parabola.

Example. Two firecrackers : one blows up 2years after the other.

1

2

3firecracker 2

firecracker 1x

t

Figure 2.3: Spacetime diagram for the two firecrackerson a Lab frame. Note that in this refer-ence frame, the two events happened at sameplace, x = 0.

Example. Previous firecrackers on a rocket framewhose speed with respect to laboratory is RL = 35

1

2

3

53

x

t

Figure 2.4: Plot of the two events on a Rocket frame mov-ing to the left. Notice that in the lab frame,the two firecrackers happened in one place.While in this rocket frame, there is a positivedirected distance from x = 0.

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Example. Previous firecrackers on a rocket framewhose speed with respect to laboratory this timeis RL = + 35

1

2

3

53

x

t

Figure 2.5: Similar to the previous example but this timethe rocket is moving to the right so the di-rected distance from the rocket to firecracker2 is to the left.

Example. A proton initially at the origin movingwith velocity 4/5 wrt lab. Plot the worldline inthe labframe.

54

x

t

Figure 2.6: Worldline of the proton

Remark. You may have noticed that in the pro-ton example, the slope is equal to 1 while in thetwo previous rocket examples, the slope is 1 .Dont be confused.

The reason the slope is positive inverse for theproton is that we were asked to plot in the labframe. In the two previous rockets, we plotted inthe rocket frame given its own relative speed to

the lab frame, RL. Note that RL = LR

2.3 Relativity of Simultaneity

Definition (Simultaneous events). Simultaneousevents are events having the same t-coordinate.

Events that are simultaneous in one IRFmay not be simultaneous in another IRF.

A special case is a set simultaneous eventshaving the same position. These will alwaysbe simultaneous.

Pairs of events occurring at the same place(x = 0) should be timelike.

Pairs of events occurring at the same time(t = 0) should be spacelike.

Example. Consider two simultaneous events, Aand B

A Bx

t

(a) Lab frame

AB

x

t

(b) Rocket frame, RL < 0

A

B

x

t

(c) Rocket frame, RL > 0

Figure 2.7: Events A and B are simultaneous in the Labframe but not in the Rocket frame.

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Remark. Suppose youre in a Lab frame and yourfriend is in a rocket frame similar to figure 2.7c.Considering what he saw in the past (event C),you just saw in the present, what defines past,present, and future? The next subsection clarifiesthe concept of causality.

2.4 Light Cones

A light cone is the path that a flash of light, em-anating from a single event (localised to a sin-gle point in space and a single moment in time)and traveling in all directions, would take throughspacetime.

future

past

t

yx

Ahyperpl

ane of the prese

nt

Figure 2.8: A light cone

Light cones plays an essential role in defining theconcept of causality, summarised below:

Event A can affect other events inside thefuture light cone.

Event A can be affected by other events in-side the past light cone

Timelike - inside the cones (can affect usinga particle)

Spacelike - outside the cones (not causallyrelated)

Lightlike - on the surface of the cone (canaffect using light)

2.5 Lorentz Transformation

Weve studied so far the geometric nature of space-time and the spacetime interval. Now we proceedwith the algebraic treatment.

Given (t, x, y, z) on a rocket frame, (t, x, y, z)on the lab frame is:

t = t + RLx (2)

x = x + RLt (3)

y = y (4)

z = z (5)

where

RL : speed of rocket wrt lab (6)

: Lorentz factor

= 11 2 (7)

For velocities,

vx =vx + 1 + vx

(8)

vy =vy

1 + vx(9)

vz =vz

1 + vx(10)

Again, use natural units for the equationsabove.

The origin of both frames are coincident.

For the inverse Lorentz transformation, re-place the primed variables with unprimedand vice versa, and note RL = LR

At non-relativistic speeds, the transforma-tion reduces to a Galilean transformation.

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2.6 Synchronisation of Clocks

Imagine spacetime to be a grid of clocks (andrulers), measuring time (and position).

Figure 2.9: Spacetime as a grid of clocks [1]. In one frameit is synchronised, in another it isnt

Synchronisation of clocks (the grid) in a referenceframe requires knowledge of the distance, D of theclock from the reference clock, usually the clockat the origin.

tset =D

c(11)

where

tset : time to set the clock

D : distance of the clock from the reference

c : speed of light

2.7 Time Dilation

Time dilation is a difference of elapsed time be-tween two events as measured by observers mov-ing relative to each other.

Clocks moving at relativistic speeds appearto run slower than a stationary clock.

Time dilation is not limited to clocks. Age-ing and other biological events dilate too!

Definition (Proper time). The time as measuredby a clock following a world line. The proper timeinterval between two events on a world line is the(unsigned) change in proper time. The propertime interval is defined by:

=

t2 x2 (12)

With the proper time defined, the formula for timedilation is:

tdilated =1 2

= (13)

Example. A House and a Church is 43L-minaway from each other. You are travelling at = +0.866 from the House to the Church. Yourbestfriend is situated in the house and is observingyour watch using a telescope. How many minuteshas elapsed in your watch (as seen by your best-friend) when his has elapsed 50min?

Using equation (12)

=

(50)2 (43.31)2

25min

Using equation (13)

= tdilated

= 501.9982

25min

2.8 Lorentz Contraction

Lorentz contraction is the phenomenon of a de-crease in length measured by the observer, of anobject which is traveling at any non-zero velocityrelative to the observer.

Objects moving at relativistic speeds appearshorter along the direction of its motioncompared to its stationary state.

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Stressing this point again, the dimensionsperpendicular to the direction of motion arenot contracted.

Definition (Proper length). It is the distance be-tween the two spacelike events, as measured in aninertial frame of reference in which the events aresimultaneous. It is given by:

=

x2 t2 (14)

The formula for length contraction is then,

L =

1 2 (15)

=

(16)

Where L : contracted length.

Example. A 1m sword is moving at = 0.866.What is the contracted length as observed by astationary alien.

L =

1 2

= (1)

1 0.8662

0.5m

2.9 Two-observer Spacetime dia-grams

We construct the rocket frame axes superimposedon the lab frame (figure 2.10).

Draw the t axis with an angle = tan1 away from from the t axis.

Draw the x axis with an angle = tan1 away from from the x axis.

If < 0, these axes lie on the 2nd and 4thquadrant respectively (same angle).

2.10 Velocity Transformation

Given an observer in a rocket frame (moving insidethat rocket) that is also moving relative to the lab

45 l

ight line

t

x

x

t

(a) RL > 0

45 l

ight line

t

x

x

t

(b) RL < 0

Figure 2.10: Two-observer spacetime diagrams

frame, the velocity of the observer with respect tolab is:

OL =OR + RL1 + ORRL

(17)

This is called the velocity-addition formula for rel-ativistic speeds.

2.10.1 The Velocity Parameter (Rapidity)

Define the s in equation (17) in terms of thehyperbolic tangent function,

tanh OR = OR

tanh RL = RL

tanh OL = OL

Equation (17) becomes

tan OL =tanh OR + tanh RL

1 + (tanh OR)(tanh RL)

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Which is similar to the relation (from Math 17):

tanh(x+ y) = tanh(x) + tanh(y)1 + tanh(x) tanh(y)

So we conclude that:

OL = OR + RL (18)

Where : velocity parameter (rapidity).

This greatly simplifies the math, as compared tothe velocity-addition formula by Equation (17).By first getting the velocity parameters in eachreference frames, we just add the two and findthe inverse hyperbolic tangent to get the relativevelocity required.

Other useful identities:

cosh RL = (19)

sinh RL = Rl (20)

Remark. If you are interested, this is hyperbolicstuff. Lol.

3 Energy, Mass, Momentum

3.1 Relativistic Doppler Effect

The apparent change in the frequency of a wavewhen there is a relative motion between the source(of the wave) and the observer.

Electromagnetic waves travel at c, so we must ac-count for relativistic effects.

3.1.1 Source moving toward the observer

fapproach = f0

1 + 1 (21)

3.1.2 Source moving away from the ob-server

frecede = f0

1 1 + (22)

Note that fapproach > f0, referred to as blueshift.

Also note that frecede < f0, referred to asred shift.

Useful identities are

f20 = fapproach frecede (23)

=

(fapproach

f0

)2 1(

fapproachf0

)2+ 1

(24)

Recall c = f

4 Relativistic Momentum

In all IRF,s the principle of conservation of mo-mentum is valid.

The generalisation of momentum is:

p = m (25)

m is rest mass.

The above equation is applied only to ob-jects with nonzero mass.

Generalising Newtons Second law, use dpdt .Not F = ma

5 Relativistic energy

In all IRFs, the work energy principle and theprinciple of conservation of energy are valid.

The generalisation of kinetic energy is:

K = ( 1)mc2 (26)

m is rest mass.

Applicable only to objects with nonzeromass.

Lorentz factor should use the velocity ofthe object.

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If velocity of the object is zero, K = 0.

As the velocity approaches infinity, K .

The generalisation of the total energy is then:

E = K +mc2

= mc2 (27)

The total energy is related to the momentum by:

E2 = (mc2)2 + (pc)2 (28)

in natural units,

E2 = m2 + p2 (29)

5.1 Invariance of Mass

From equation (29),

m2 = E2 p2

This mass is invariant, so for any frame of refer-ence, we have the equation

E2 p2 = (E)2 (p)2 (30)

References

[1] Tatsu Takechi. Synchronization of clocks. on-line.

[2] Wikipedia. Wikipedia, the free encyclopedia,2004.

[3] H.D. Young, R.A. Freedman, and A.L. Ford.Sears and Zemanskys University Physics:With Modern Physics. Addison-Wesley, 2012.

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