Physics 73 notes

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

UP

K E M

• No change in speed or direction (non-accelerating frame). • No gravitational force is felt (Earth cannot be a true IRF, but is approximated as such)

1.2

Newtonian Relativity

• Laws of mechanics are preserved in all IRF’s. • This is familiar to you already, recall relative velocities from Physics 71.

MEMBERSHIP ACADEMIC DEVELOPMENT

• Invalid for speeds near the speed of light.

Physics 73 1.3

2nd Long Exam Reviewer

1.3.1

Heavily based on Arciaga Notes

Special Relativity First Postulate

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

Preface 1.3.2 This handout is intended as a reviewer only and should not be substituted for a complete lecture, or used as a reference material. The goal of this reviewer is to refresh the student on the concepts and techniques in one reading. But this is more than enough to replace your notes :)

1

Second Postulate

The speed of light in a vacuum is constant in all inertial frame of reference and is independent of the motion of the source. Examples. Fixed quantities • Numerical value of the speed of light in a vacuum

Principle of Relativity

• Value of the charge on the electron

1.1

Reference Frames

• Order of the elements in the Periodic Table • Newton’s First Law of Motion

Definition (Reference frame). A reference frame is simply a coordinate system attached to a particular observer.

Examples. Examples of variable quantities

Definition (Inertial frame of reference). An inertial frame of reference is a reference frame in which Newton’s 1st law is valid

• Speed • Time between two events

1

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

2.1

• Kinetic energy

Invariance of the Interval

• Force Definition (Spacetime Interval). The interval between two events happening at some points (ta , xa ) and (tb , xb ). The interval, ∆s, is defined by

• Electric field • Magnetic field

∆s2 = ∆t2 − ∆x2 = (∆t0 )2 − (∆x0 )2

1.4

Natural Units

(1)

∆x = xb − xa

To elegantly simplify particular algebraic expressions appearing in the laws of physics (particularly relativity here), we deal with natural units. • Same unit for space and time

∆t = tb − ta With the 0 denoting an observation made in another frame of reference. Spacetime interval is invariant, i.e. observations in two different IRF’s has the same spacetime interval.

• Conversion factor : c = 299 798 482 m/s

Table 2.1: Comparison of two spaces

• In natural units, c = 1 vnat =

Flat spacetime

1X Xnat = tnat c t

Places

(t, x, y, z)

(x, y, z)

invariant interval

invariant distance

∆s2 = c2 ∆t2 − ∆x2

7

Euclidean space

Events tnat = ct • Speed is unitless

vs




∆d2 = ∆x2 + ∆y 2




Example. Convert 6.21 × 10 m/s to natural units 6.21 × 1017 m/s = 0.207 3 × 108 m/s

2.2

Example. Convert 20 min to meters   60 s  m 20 min 3 × 108 = 3.6 × 1011 m 1 min s

Spacetime Diagrams

A spacetime diagram is a helpful visual representation of events in spacetime. Note the following: • In one spatial dimension, the vertical axis is t while the horizontal is x.

Remark. From here on, all quantities will be in natural units.

2

Events and Measurements

• An event is a point, with coordinates (t, x). • A worldline is a curve on the diagram. This represents the history of events (refer to figure 2.1). • Classifications of spacetime intervals:

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

– Timelike : ∆t2 − ∆x2 > 0

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

– Light-like : ∆t2 − ∆x2 = 0

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– Spacellike : ∆t2 − ∆x2 < 0

University of the Philippines Chemical Engineering Society, Inc. (UP KEM) Physics 73 - 2nd Long Exam Example. Two firecrackers : one blows up 2 years after the other.

event

t

t 3 firecracker 2 2 worldline 1 firecracker 1

x

x Figure 2.1: An example of a spacetime diagram

• No particle can travel at and exceed the speed of light light. That is: v 0 Figure 2.7: Events A and B are simultaneous in the Lab frame but not in the Rocket frame.

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University of the Philippines Chemical Engineering Society, Inc. (UP KEM) Physics 73 - 2nd Long Exam Remark. Suppose you’re in a Lab frame and your friend 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 clarifies the concept of causality.

2.4

• Lightlike - on the surface of the cone (can affect using light)

2.5

Lorentz Transformation

We’ve studied so far the geometric nature of spacetime and the spacetime interval. Now we proceed with the algebraic treatment.

Light Cones

A light cone is the path that a flash of light, emanating from a single event (localised to a single point in space and a single moment in time) and traveling in all directions, would take through spacetime.

Given (t0 , x0 , y 0 , z 0 ) on a rocket frame, (t, x, y, z) on the lab frame is: t = γt0 + γβRL x0

(2)

x = γx0 + γβRL t0

(3)

t

y = y0

(4)

future

z = z0

(5)

where βRL : speed of rocket wrt lab A

x

sent he pre

of t plane r e p y h

(6)

γ : Lorentz factor 1 =p 1 − β2

y

(7)

For velocities,

past

Figure 2.8: A light cone

Light cones plays an essential role in defining the concept of causality, summarised below: • Event A can affect other events inside the future light cone. • Event A can be affected by other events inside the past light cone • Timelike - inside the cones (can affect using a particle) • Spacelike - outside the cones (not causally related)

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vx =

vx0 + β 1 + βvx0

(8)

vy =

vy0 1 + βvx0

(9)

vz =

vz0 1 + βvx0

(10)

• Again, use natural units for the equations above. • The origin of both frames are coincident. • For the inverse Lorentz transformation, replace the primed variables with unprimed and vice versa, and note βRL = −βLR • At non-relativistic speeds, the transformation reduces to a Galilean transformation.

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

2.6

Synchronisation of Clocks

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

Definition (Proper time). The time as measured by a clock following a world line. The proper time interval between two events on a world line is the (unsigned) change in proper time. The proper time interval is defined by: p ∆τ = ∆t2 − ∆x2 (12)

With the proper time defined, the formula for time dilation is: ∆tdilated = p

∆τ 1 − β2

= γ∆τ

Figure 2.9: Spacetime as a grid of clocks [1]. In one frame it is synchronised, in another it isn’t

Synchronisation of clocks (the grid) in a reference frame requires knowledge of the distance, D of the clock from the reference clock, usually the clock at the origin. D (11) tset = c where

(13)

Example. A House and a Church is 43 L-min away from each other. You are travelling at β = +0.866 from the House to the Church. Your bestfriend is situated in the house and is observing your watch using a telescope. How many minutes has elapsed in your watch (as seen by your bestfriend) when his has elapsed 50 min? Using equation (12) p ∆τ = (50)2 − (43.31)2 ≈ 25 min Using equation (13)

tset : time to set the clock

∆τ =

D : distance of the clock from the reference = c : speed of light

∆tdilated γ 50 1.9982

≈ 25 min

2.7

Time Dilation 2.8

Time dilation is a difference of elapsed time between two events as measured by observers moving relative to each other. • Clocks moving at relativistic speeds appear to run slower than a stationary clock.

Lorentz Contraction

Lorentz contraction is the phenomenon of a decrease in length measured by the observer, of an object which is traveling at any non-zero velocity relative to the observer.

• Time dilation is not limited to clocks. Ageing and other biological events dilate too!

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• Objects moving at relativistic speeds appear shorter along the direction of its motion compared to its stationary state.

University of the Philippines Chemical Engineering Society, Inc. (UP KEM) Physics 73 - 2nd Long Exam • Stressing this point again, the dimensions perpendicular to the direction of motion are not contracted.

t0 45 ◦ “l ig ht

lin

e”

t

Definition (Proper length). It is the distance between the two spacelike events, as measured in an inertial frame of reference in which the events are simultaneous. It is given by: p (14) ∆σ = ∆x2 − ∆t2

x0

α x

α

The formula for length contraction is then, p

1 − β2

(a) βRL > 0

(15)

t ”

t

∆σ = γ

0

lin e

(16)

45 ◦ “l ig ht

L = ∆σ

Where L : contracted length. Example. A 1 m sword is moving at β = 0.866. What is the contracted length as observed by a stationary alien. L = ∆σ

p

α x α

1 − β2

p = (1) 1 − 0.8662

x0

≈ 0.5 m

(b) βRL < 0 Figure 2.10: Two-observer spacetime diagrams

2.9

Two-observer Spacetime diagrams

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

frame, the velocity of the observer with respect to lab is: βOR + βRL βOL = (17) 1 + βOR βRL This is called the velocity-addition formula for relativistic speeds.

• Draw the t0 axis with an angle α = tan−1 β away from from the t axis. • Draw the x0 axis with an angle α = tan−1 β away from from the x axis. • If β < 0, these axes lie on the 2nd and 4th quadrant respectively (same angle).

2.10.1

The Velocity Parameter (Rapidity)

Define the β’s in equation (17) in terms of the hyperbolic tangent function, tanh θOR = βOR tanh θRL = βRL

2.10

Velocity Transformation

tanh θOL = βOL

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

Equation (17) becomes

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tan θOL =

tanh θOR + tanh θRL 1 + (tanh θOR )(tanh θRL )

University of the Philippines Chemical Engineering Society, Inc. (UP KEM) Physics 73 - 2nd Long Exam • Note that fapproach > f0 , referred to as “blue shift”.

Which is similar to the relation (from Math 17): tanh(x + y) =

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

• Also note that frecede < f0 , referred to as “red shift”.

So we conclude that: θOL = θOR + θRL

• Useful identities are

(18)

f02 = fapproach · frecede

Where θ : velocity parameter (rapidity).

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

β=

cosh θRL = γ sinh θRL = γβRl

fapproach f0

2 2

−1 (24) +1

• Recall c = γf

4

Other useful identities:

fapproach f0

Relativistic Momentum

(19) (20)

Remark. If you are interested, this is hyperbolic stuff. Lol.

In all IRF,s the principle of conservation of momentum is valid. The generalisation of momentum is: p = γmβ

3 3.1

(25)

• m is rest mass.

Energy, Mass, Momentum

• The above equation is applied only to objects with nonzero mass.

Relativistic Doppler Effect

dp . • Generalising Newton’s Second law, use dt Not F = ma

The apparent change in the frequency of a wave when there is a relative motion between the source (of the wave) and the observer. Electromagnetic waves travel at c, so we must account for relativistic effects.

3.1.1

(23)

Source moving toward the observer

5

Relativistic energy

In all IRF’s, the work energy principle and the principle of conservation of energy are valid. The generalisation of kinetic energy is:

s fapproach = f0

3.1.2

1+β 1−β

(21)

Source moving away from the observer s frecede = f0

1−β 1+β

(22)

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K = (γ − 1)mc2

(26)

• m is rest mass. • Applicable only to objects with nonzero mass. • Lorentz factor γ should use the velocity of the object.

University of the Philippines Chemical Engineering Society, Inc. (UP KEM) Physics 73 - 2nd Long Exam • 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 = γmc2

(27)

The total energy is related to the momentum by: E 2 = (mc2 )2 + (pc)2

(28)

in natural units, E 2 = m2 + p2

5.1

(29)

Invariance of Mass

From equation (29), m2 = E 2 − p2 This mass is invariant, so for any frame of reference, we have the equation E 2 − p2 = (E 0 )2 − (p0 )2

(30)

References [1] Tatsu Takechi. Synchronization of clocks. online. [2] Wikipedia. Wikipedia, the free encyclopedia, 2004. [3] H.D. Young, R.A. Freedman, and A.L. Ford. Sears and Zemansky’s University Physics: With Modern Physics. Addison-Wesley, 2012.

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