Week 9 Flashcards
What are the postulates of special relativity?
- Principle of relativity - laws of physics apply equally on all inertial frames
- Universal speed of light: C in a vacuum is the same for all internal observer, regardless of motion of source or observer
What is Einsteins velocity addition rule? (I’m walking A, B is the train and the ground is C)
V ac = ( V ab + V bc ) / (1 + ( V ab V bc /c^2))
What is relativity of simutatineit
Two events that are simultaneous in one inertial system are no in general simultaneous in another
What is Lorentz contraction?
Objects moving at high speed are shorted in direction of motion compared to when they are at rest
Lorentz length equations
L moving = Lrest/ Gamma
What is time dilation I?
Time at high speed run sow compare to at rest
What is time dilation equation?
Time moving = Time rest * gamma
What does the overbar represent?
The moving frame
How to convert proper length to moving?
X bar = Gamma ( x -vt)
How to convert proper time to moving?
T bar = gamma ( t- vx/ (C^2) )
What is beta?
V/c
What would give an invariant quantity?
The scalar product between 4-vectors
What shape is cause by the 4-vector rotation?
Hyperboloid due to - sign
Whydoes the 4d hyperboloid shape make sense?
Because energy-momentum and 4d scalar equations are both hyperboloid equations
What does it mean if I > 0?
Interval is space like and not causally connected as no light can travel fast enough to connect them and they are simultaneous for some observer
What does it mean if I<0?
It means it is time-like and causally connected so one event can influence the other, may not be simulatnaoues
What is I = ?
Light like, light travelling fast enough from one could reach the other. Always simultaneous
What is rapidity?
A measure of relative velocity, theta
Tan theta = relative velocity/ c
What are contravariant 4 vectors?
Contravariant 4-vectors have upper indices and transform with the Lorentz transformation matrix
Λ
ν
μ
Λ
ν
μ
. They describe quantities like position and momentum that “act” on spacetime.
What are covariant 4 vectors?
Covariant 4-vectors have lower indices and transform with the inverse Lorentz transformation matrix
Λ
μ
ν
Λ
μ
ν
. They describe quantities like gradients and fluxes that “respond” to spacetime.
Define proper time
Time measured from point of view of moving object (ie moving frame in which object in stationary)
What is the equation for energy-momentum 4 vector?
P Sup mu = M n sup MU
Formula for relativistic momentum
P = mn = γmu
Formulae for relativistic energy
E = γmc^2 = Erest + E kinetic
Formula for Erest?
Mc^2
Formula for Ekinetic
(γ - 1)mc^2
What is ordinary speed?
U, distance/time
- used for non-relativistic
How can proper velocity be >c ?
Depends on reference frame.
Does not violate relativity as proper velocity is not the same as ordinary speed
Which reference frame is ordinary velocity measured in?
- The inertial reference frame OR
- relative to a stationary object or a moving observer
Relativistic energy and momentum relation derivation
- 4 vectors
- p sup mu = (E/c, P)
- Minkowski Metric
- S^2 = -(ct)^2 + x^2 + y^2 +^2
-> s^2 = - (E/c)^2 +p^2
- In restframe where p-0 and E =mc^2
S sup mu = (Eo/c , 0)
P sup mu p sub mu = ( (mc^2)/c )^2 = mc^2
- So for any frame
- E^2/C^2 - p^2 = m^2c^2
What is mostly-plus?
3 of 4 vectors are +ve
- high energy/general relativity
- Causality- Spacetime intervals to ensure that timelike intervals are negative and space like are positive
What is mostly-minus?
3 of 4 vectors are -ve
- Quantum and theoretical
Which reference frames are Maxwell’s equations right in?
All due to c
How does EM transfer across reference frames?
E and B depend on reference frames but Lorentz force(and therefore E and P ) are invariant
How is Lorentz invariant when E + B are not?
- Lorentz force is an invariant EM field tensor
Which velocity is used in Lorentz force?
The velocity in the reference frame of the observer who is applying the force equation
What does the experiment of moving wires charges show?
Problem of length contraction can only be solved by magnetic field
the existence of magnetic force follows, as a logical necessity, from the existence of an electrical force together with the principe of relativity
How are electric force transformed?
How does time dilation affect charge and current density?
- Charge frame is invariant
- Current is affected - dilated in stationary
How does length contraction affect charge and current density?
The moving observer sees charge density increased in the direction of moving
How are E and B fields related by relativity?
B is an E field in another reference frame
Explain simply a solenoid moving along its axis with regards to electromagnetic relativity
- Rest frame: uniform field
- Moving frame: magnetic field perpendicular to motion can be transformed into additional fields -> new E field
How does length contraction affect a solenoid moving along its axis with regards to electromagnetic relativity
The number of turn per unit length is denser
( n bar = γn)
How does time dilation affect a solenoid moving along its axis with regards to electromagnetic relativity
The solenoids clock runs slower, so from Sbar (moving), the current in solenoid (charge/time) must be smaller
I bar = I/γ
How does relativity affect a moving solenoid?
- Lower current
- Increased turn density
- Effects cancel -> effect in direction of movement does not change between reference frames
Explain a capacitor moving perpendicular to its plates in terms of electromagnetic relativity
- Distance between plates length contracted
- Plates infinitely thin so ultimately unaffected
- Charge density unaffected
- E field does not change between reference frames
- B field in moving frame as charges now moving relative to observer
What does length contraction look like formulaically?
Moving length = rest length/gamma
Describe a capacitor moving parallel to its plates in terms of electromagnetic relativity
- E and B fields mix so need to consider both simultaneously in both frames
- Consider rest frame and frames where each at rest relative to rest frame and each other
What happens in relativity with field components perpendicular to direction of motion?
E and B fields mix -> need 2 frames with
How to transform electric fields between frames S and S bar moving at speed v along direction x-
X: E bar = E
Y: E bar = γ ( Ey - vBz)
Z: E bar = γ(Ez + vBy)
How to transform magnetic fields between frames S and S bar moving at speed v along direction x-
X: B bar = B
Y: By bar = γ ( By + ( v/ c^2 ) Ez)
Z:Bz bar = γ (Bz -( v / c^2) Ey)
Explain simply the benefit and rationale of a 2-rank 4-tensor over a 4-vector
Used to describe physical quantities that are more complex because the are connected e.g. E/B or energy-momentum
- has 2 indices
Why can’t we put E and B into a tensor?
They have 6 components total which mix with each other under a Lorentz boost
What is a Lorentz boost?
Transformation used to describe how coordinates change when you change reference frames
What is the difference between 4vecctor and rank-2 4-vector?
Latter has 2 indices instead of one
What are symmetric tensors?
- t mu v = t v mu
- Represents physical quantities involving scalar relationship that are without direction (energy/momentum)
What are antisymmetric tensors?
- t Mu v = - t v Mu -> t mumu = 0 so diagonal. = 0
- Represent physical quantities with directional properties (E/B fields)
What is a dual tensor?
A new tensor created from an original tensor to represent a different perspective on the same physical situation.
What does the EM dual tensor do?
Describes the same E and B fields but highlighting the relationship between the fields
What is the duality symmetry of electromagnetism?
- E and B fields can transform into each other under certain conditions
- I.E. if swap E and B fields and adjust their strengths in a specific way, EM laws e.g. MAxwell equations, still hold true
What are some consequences of the duality symmetry of electromagnetism?
- E and B fields are interchangeable under certain conditions
- Possibility of magnetic monopoles
- Maxwells equations are symmetric
What are the components of the current density 4-vector?
C Rho (charge density) and 3 spatial vectors
How does the current density change due to relativity when cloud of Q moves with velocity ?
- In rest frame, volume if larger by gamma so charge density is lower by gamma
What is proper charge density?
The charge density measured in the frame that moves together with the charge
What is current density equation for cloud Q moving at velocity v?
J = Gamma Rho o U
What is charge density equation for cloud Q moving at velocity v?
Rho = gamma * Rho o (proper charge density)
What is the continuity condition in 4-vector notation and what does it state?
(Delta J sup mu / (delta x sup mu) = 0
Divergence-less in al 4 dimensions
What do the indices in tensors represent?
1st rank = refers to the component of the vector in a given direction
2nd rank = each corresponding to a direction in spacetime
What does the current density 4-vector do?
Unifies the charge and current density in one spacetime framework
What represents charge and current in spacetime?
Oth component = c Rho = charge density
Ith component = 1,2,3 = current density (flow of charge in space)
What is the Minkowski force?
Relativistic version of classical force
- rate of change of 4-momentum/ change in proper times
What is noted about the proper velocity in Minkowski force?
V is subscript -> covariant form
- temporal component has a negative sign
What are the components of Minkowski force?
- Spatial = rate of change in momentum in space
- Time = rate of change of energy wrt proper times (how object’s energy changes in response to external forces)
What is proper velocity?
Distance covered by the moving object (dl) measured in the observer’s frame,per unit proper time dTau (measured in object’s rest frame)
N = dl/dt
