Chapter 37: Special Relativity

Question 1

Inertial reference frame

Answer 1

• a reference frame in which NI holds.
• inertial reference frames are either at rest or move with a constant velocity
• any two such inertial refernce frames move at a constant velocity relative to eahc other

Question 2

non-inertial reference frame

Answer 2

A reference frame which is accelerating either in a linear fashion or rotating about some axis (centripital acceleration)

Question 3

Event

Answer 3

-An occurrence at a definite instant in time and point in space. By using a reference
frame we can assign space and time coordinates to an event.
For example: A light flashes/a balloon pops/two particles collide at position (x, y, z) and time t.

Question 4

observer

Answer 4

A (possibly hypothetical) person who performs measurements in a reference frame in which they are stationary. They interpret the results of these measurements intelligently.

Question 5

Einstein’s Postulates of Relativity: 2

Answer 5

(a) The laws of physics are the same in all inertial reference frames. (this is simply an extenmtion to the more general principle of relativity from before)

(b) The speed of light (in vacuum) is the same in all inertial frames, regardless of the motion of the source.
i. e. light travels thorugh empty spoace with a constant velocity c that is independent of the speed of observer or source of light…

thus, acc the MM experiment this theory of special relativity rejects the existance of an ether

Question 6

state the priciple of relactivity

Answer 6

the basic laws of physics are the same in all inertial reference frames

from one reference frame to another, thibngs like forces, mass, length and time does not change, these quantiteis are said to be absolute

Question 7

since we have the [principle of relativity and since this means that the laws of mechanics do not chnage for differnet inertial refernece frames what does this mean

Answer 7

no one inerital refernce frame is special in any sense

we can conclude that all inertial reference frames are equivalent

Question 8

when are to events cosnidered to be simultaneous

Answer 8

when they happen at the same time

they may have i=different positions in space as per the def of an event

Question 9

what is wrong with the galilean transform

Answer 9

it is at best a low speed ( non-relativistic ) approximation of a more general transformation law ( the lorentz transform

it is not compatible with einstein’s second postulate

Question 10

the relativity of simultaneity

Answer 10

events that occur simultaneously in one frame need not occur simultaneously in other frames.

there is no ‘right’ or ‘wrong’ here. “now” simply means different things to different observers

all three ref frmes in our eg (rocket, cart, ground) are equally valid and on the same footing

Question 11

why does the relativity of simulteiniety not spell trouble for causality, the principle that cause alwyas precedes effect

Answer 11

no
if one event can cause/ in any way influence another evnet then the order of the two will be the same in all frames.

Although the time difference (interval) between the events can be different in different frames

Question 12

what is time dilation

Answer 12

moving clocks run slower than stationary clocks

Question 13

what is the formula for the lorentz factor in words and eq

Answer 13

the factor by which time, length, and relativistic mass change for an object while that object is moving.

λ = 1/√1-(v^2/c^2)

Question 14

why don’t we notice time dilation in classical mechanics where things are large and slow

Answer 14

v^2/c^2 tends to 0
λ = 1/√1-(v^2/c^2) tends to 1
the lorentz factor λ tends to 1 as v tends 0

if we put an atomic clockc ona plane it would indeed measure nanoseconds slower so it does happen in classical terms just we have to ‘‘zoooooom in’’

Question 15

why do we notice time dilation so much only at relativistic speeds

Answer 15

the lorentz factor λ tends to ∞ as v tends to c

Question 16

give time dilation formula

Answer 16

Δt = λ Δto

more time passes between two events in a movign ref frame than a stationary one .
thsu we have that movign clocks run slower than stationary clocks

NB: only works if Δto is the proper time (the shortest possible time betwene the events)

Question 17

proper time between two events

Answer 17

the time measured in the frame inwhich the two events occur at the same point in space

this is the shortest possible time betwene the two events

Question 18

to an observer in S, a clock which is stationary in S’ will run slow by a factor of λ. if the clock ticks once every second in S’ then how often does it tick in S?

Answer 18

it will tick once every λ>1 seconds in S

S’ is the home frame
S, according to time dilation will have time that runs slower as it is the moving frame and Δt = λ Δto and gamma is never negative

Question 19

Discuss what is time dilation

Answer 19

since Δt = λ Δto and thus that more time passes berween events ina moving frmae than a stationary one and so it appears that we have an apparent ‘stretching of time’

Question 20

If we have 2 inertial ref frames. R and R’ and λ = 2. R’ is moving wrt R at a speed of u

the person in R says: “ my clock ticks every second but yours ticks every two seconds. you clock is slow”

what does the pereosn in R’ say??

Answer 20

ditto: to this obeserver in R’ he himself is stationary and thus the guy in R is moving at a velocity of -u away from him.

so the same and the person in R’ says: “ my clock ticks every second but yours ticks every two seconds. you clock is slow”

Question 21

length contraction eq

Answer 21

l= lo/c

Question 22

length contraction in words

Answer 22

the length of a moving object, along its direction of motion, it always SHORTER than its proper length

lengths along directions perpendicular to that of the motion are not modified

Question 23

proper length

Answer 23

the length of an object in its rest frame

Question 24

in a moving ref frame an person/object A will see that other lengths are ………..
while from another ref frame the object A will be………..

Answer 24

contracted and shortened along its direction bya factor of length contraction

contracted in a similar way along the direction of motion only and by a factor of gamma

Question 25

lsit all the relativistic quantities

Answer 25

time
length
energy
momentum
even velocity f you think of dr/dt'

Question 26

what is relativistic velcoity

Answer 26

dr/dt’

at low speeds t = t’ and becomes dr/dt

Question 27

relativistic momentum in words and eq

Answer 27

p = mv(relativistic) = m(dr/dt') = m (dr/dt)(dt/dt') = m λ v
p = mv/√1-(v^2/c^2)

this momentum is conserved in all ref frames

Question 28

4 vector

Answer 28

(t,x,y,z)
spacetime
spacetime is any mathematical model which fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. known as minkowski space

Question 29

is memontum in the time direction Pt conserved

Answer 29

yes, up to a constant facctor, the clock’s relativistic energy
E= Pt c^2 = m λ c^2

Question 30

Rest mass energy

Answer 30

E= mc^2
this mathematically relates mass and energy.
implies that mass is simply a form of energy

Question 31

there is an upper limit on v , why is there no upper limit on E or p

Answer 31

E = m λ c^2 or p=m λ have no lkmit
this is due to the the λ factor converging as v approches c. lambda approaches ∞ as v tends to c. then n both E and p approach ∞ also

Question 32

are total relativistic energy and momentum conserved

Answer 32

yes, but the total rest mass energy and kintetic energy are NOT cosnerved individually. There ae re processes that convert the one to the other and vice versa

Question 33

describe how we get to E^2 = (pc)^2 + (mc^2)^2 eq

Answer 33

p = λ mv, take m tends to 0 and v tends to c st p remains fixed. Then E^2 = (pc)^2 + (mc^2)^2 eq beocmes E=pc. this is means that the particle is massless and has energy pc which moves at the speed of light. this is a photon