- in the late 19th century it was though that a wave needed a medium to propagate through - the medium though to support electromagnetic radiation was called the aether

Special Relativity Flashcards by Sophie Wilkinson

The Aether

in the late 19th century it was though that a wave needed a medium to propagate through
the medium though to support electromagnetic radiation was called the aether

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The Michelson-Morley Experiment

Purpose

-the experiment was designed to measure the velocity of the aether

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The Michelson-Morley Experiment

Equipment

light from a source was directed horizontally towards an angled half silvered mirror
half of the light reflected upwards and half passed straight through the mirror
both of these beams of light travel towards fully silvered mirror a distance L away

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The Michelson-Morley Experiment

Rest Frame

assumes that the velocity of the ‘aether’ is 0

- both beams of light travel the same distance and at the end they are in phase

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The Michelson-Morley Experiment

Moving Frame

assumes that the light is moving through a medium, the aether, which itself is moving at a velocity, v
this has no effect on the horizontal light beam as it is moving in the same direction as the aether
this means that the vertical beam appears to travel further so is out of phase with the other beam at the end of the experiment

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The Michelson-Morley Experiment

Equations

-the ratio between the time taken for the light to take the transverse path and the time taken to take the longitudinal is:
Tl / Tt = 1 / √(1 - (v/c)²)
-this was the prediction of the physics available at the time
-but after many attempts, no measured difference could be found
-they concluded that the aether does not exist

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Fitzgerald-Lorentz Contraction

-Heaviside calculated that the electric field surrounding a spherical distribution of charge should cease to have spherical symmetry once the charge is in motion
-Fitzgerald suggested that this distorted charge cloud applied to macroscopic objects, the molecular field accounts for the Michelson-Morley result
-Fitzgerald proposes that the arm of the apparatus extended in the direction of motion is contracted by an amount proportional to
√(1 - (v/c)²)

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Lorentz Transforms

First Equation - Distance

-suppose a distance x’ is measured in S’, the moving frame
-but in S, the stationary frame, the rule x’ is contracted to x’√(1 - (v/c)²)
-if the frame S’ has travelled a distance vt
-then the endpoint will be measured to be
x = x’√(1 - (v/c)²) + vt

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Reference Frame

Definition

a set of coordinate systems at rest relative to each other is called a reference frame

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The Galilean Transformations of Position

x = x' + vt
y = y'
z = z'

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The Galilean Transforations of Velocity

Ux = Ux' + V
Uy = Uy'
Uz = Uz'

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Inertial Reference Frames

Definition

reference frames that are not accelerating

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Galilean / Newtonian Principle of Relativity

if we have two inertial reference frames moving with constant velocity relative to each other, there are no mechanics experiments that can tell us which is at rest and which is moving or if they both are moving

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Electrodynamics - The Problem

if you transform Maxwell’s equations using Galilean transforms then their form is different in different reference frames
this implies that we can use electrical or optical experiments to determine the speed of the references frame
Maxwell’s equations state that the velocity of light is a constant, even if the source is moving
BUT, Galilean transforms predict that c is not constant, if we were moving at u, light would pass us at, c-u

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Constant of the Speed pf Light

all the laws of physics are the same in all inertial reference frames
Maxwell’s equations are true in all inertial reference frames
Maxwell’s equations predict the speed of light in a vacuum is c
therefore light travels at the same speed c in all inertial reference frames

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The Two Postulates of Special Relativity

no experiment can determine whether you are at rest or moving uniformly
the speed of light is independent of the speed of the light source

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Time Dilation

t = γt’

times passes slower for a moving object

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Gamma

γ = 1 / √(1 - (v/c)²)

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Simultaneity

Definition

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two events that take place at different positions, (x1 and x2) but at the same time (t1=t2) as measured is some reference frame are said to be simultaneous, IN THAT REFERENCE FRAME

The Lorentz Transformations

S’

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x' = γ (x - vt)
y' = y
z' = z
t' = γ (t - vx/c²)

The Lorentz Transformations

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x = γ (x' + vt)
y = y'
z = z'
t = γ (t' +vx'/c²)

What are the Lorentz Transformations?

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transformations that allow Maxwell’s equations to retain their form
all physical laws should remain unchanged under the Lorentz transformations

Length Contraction

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L = L’/γ

where L’ is the length measured in the rest frame of the object

How do you find the time dilation formula?

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using the time transformation when the distance is zero

How do you find the length contraction formula?

using the distance transformation when the time is zero

Lorentz Velocity Transformations | x

Ux' = (Ux-v) / (1 - Ux*v/c²)

Lorentz Velocity Transformations | y

Uy' = Uy / γ(1 - Ux*v/c²)

Lorentz Velocity Transformations | z

Uz' == Uz / γ(1 - Ux*v/c²)

Lorentz Velocity Transformations | v << c

-if v< 1 and (1 - Ux*v/c²) -> 1 -so for v<

Relativistic Momentum

ρ = m0*γ*u | -this clearly reduced to the Newtonian momentum, ρ=m0*u for u<

Relativistic Energy | Equation

E² = ρ²c² + (m0c²)²

Deriving the Relativistic Kinetic Energy Equation

``` -using the binomial expansion: γ*m0 = m0 (1 + u²/2c²) -> ρ = mu = γ*m0*u -> m = γ*m0 m ≈ m0*(1 + u²/2c²) =m0+ m0*u²/2c² ->mc² ≈ m0c² +m0*u²/2 -> KE ≈ mc² - m0c² = c² (m - m0) =m0c² (γ - 1) ```

Relativistic Kinetic Energy Equation

KE = m0c² (γ - 1)

Deriving the Relativistic Energy Equations

``` -start with: dE/dt = F*u d(mc²)/dt = u*d(mu)/dt -multiply both sides by 2m c²*2m*dm/dt = 2mu*d(mu)/dt c²*d(m²)/dt = d(mu)²/dt -if two derivatives are equal to each other then the quantities differ at most by a constant m²c² = (mu)² + C -when u=0, m=m0 : m0²c² = 0 + C C = m0²c² m²c² = (mu)² + m0²c² -multiply everything by c² m²c^4 = (mu)²c² + m0²c^4 -substitute ρ=mu and E=mc² E = ρ²c² + (m0c²)² ```

Cartesian Invariant

-the two coordinate systems C and C' are identical except for their orientation and origins -suppose the endpoints of an object d are measured in both C and C' -in C the end points are recorded as (x1,y1) and (x2,y2) -in C' the end points are recorded as (x1',y1') and (x2',y2') -none of these measurements agree, even the intervals Δx≠Δx' and Δy≠Δy' -however the following can be agreed on independent of the coordinate system used d²=(Δx)² + (Δy)²=(Δx')² + (Δy')²

What is the Cartesian invariant an invariant?

it has the same value in any coordinate system

Space Time Interval

``` s² = c²t² - x² - y² - z² s² = c²(Δt)² - (Δx)² ``` - s has the same value in all inertial reference frames - it is a relativistic invariant

Four Vectors

- sets of four coordinates that consist of pairs of quantities that transform the same way under the Lorentz transformations e. g. (x,y,z,t) and (ρx, ρy, ρz, E)

Evidence for the Neutrino

-the kinetic energy of the electron varies which implies that their must be another particle that shares the kinetic energy

Maximum Kinetic Energy of Electron from Neutron Decay

-leave out the neutrino and share all of the energy between the electron and the proton mnc² = Ep + Ee mnc² = mpc²+mec²+KEp+KEe -> KEp+KEe = c²(mn-mp-me) KEp + KEe = 0.783MeV -this larger than the rest mass energy of the electron so we definitely need relativistic equations -using conservation of momentum, ρp = -ρe COME BACK TO THIS

Momentum, Energy and Light

Momentum and energy are equivalent for light (in c=1 units), as photons have no mass E² = ρ²c² + (m0c²)² E² - ρ²c² = 0 E = ρc

Relativistic Doppler Shift | Description

- in the rest frame, S, of a laser we can count how many wavefronts leave per second, f, and the distance between these wave fronts, λ - but now consider a moving frame, S' - the time in S' is dilated so t'=γt, where t is the proper time measured in S - but t' is not the observed time between the wavefronts in S' as the observer also moves a distance vt' in this time - the wavefront travelling at c takes vt'/c longer to reach the observer

Relativistic Doppler Shift | Frequency Equation

f' = f*√{(1 - v/c) / (1 + v/c)}

Relativistic Doppler Shift | Source and Observer Approaching Each Other

-frequency observed is higher, blueshifted

Relativistic Doppler Shift | Source and Observer Receding From Each Other

-frequency observed is lower, redshifted

Relativistic Doppler Shift | Alternative Frequency Equation

f' = γ*(f - v/λ) - remember that k=1/λ - this is another example of a pair of quantities, frequency and wavevector that transform with the Lorentz transformations

Special Relativity Flashcards

(46 cards)