Radiation and Relativity Flashcards

Flashcards on Radiation and Relativity

What is the key idea behind a covariant approach in physics?

Physical laws should look the same in every valid reference frame, and no reference frame is privileged.

What mathematical objects emerge when ensuring consistency under transformation between reference frames?

Tensors, in classical Newtonian mechanics, electromagnetism, and special and general relativity.

In classical mechanics, what describes the motion of a moving point particle?

A 3-vector r(t) with spatial coordinates x, y, z parametrized by time t.

How many coordinates do points of spacetime have?

Four coordinates: (t, x, y, z).

What role do inertial frames play in special relativity and classical mechanics?

A special role; they are the counterparts of Cartesian coordinates in 2D and 3D space.

If a point particle experiences no force, what does Newton’s first law state?

If it is at rest it will stay at rest, or if it is in uniform motion it will stay in that motion.

When Σ and Σ are in standard configuration, what 3 assumptions are made?

When t = 0, then t = 0 and the spatial axes are aligned, Σ travels at +v along the x-axis, the spacetime origins in both frames agree.

What is the key feature of Galilean transformations regarding time?

In all inertial frames, all inertial observers agree that time is universal.

What discovery led Maxwell to propose that light is an electromagnetic phenomenon?

Electric and magnetic fields in free space satisfy d’Alembert’s equation, i.e. the time-dependent wave equation.

What is the d'Alembert's equation?

1/c^2 * (d^2f/dt^2) - ∇^2f = 0

Who proposed that lengths contract to explain the Michelson-Morley observations?

George Fitzgerald and Hendrik Lorentz.

Who said, “…the problems are resolved when we redefine transformations between inertial frames via Lorentz transformations rather than Galilean transformations?”

Albert Einstein.

What are Einstein’s two postulates of special relativity?

E1. The laws of physics have the same form in all inertial reference frames. E2. Light propagates through empty space with speed c, independent of the speed of the observer and source.

What is one metre of ct equivalent to?

About 3.3 ns of time.

What is rapidity?

w ≡ arctanh v/c, as a dimensionless parameterisation of velocity with w → ±∞ as v → ±c

In the Lorentz transformation matrix, what do partial derivatives imply?

The other ‘new’ variables are constant.

What is the light cone of an event?

The set of all the points on all light rays through an event.

What is proper time?

Time as measured by a stationary clock.

What is Minkowski space?

The geometric structure of relativistic spacetime.

What is the Euclidean metric?

gij = ei · ej = δij

How do physical equations need to transform to be covariant?

Physical equations should be covariant/transform covariantly.

What do Roman and Greek letters represent in Einstein's convention?

Roman letters represent spatial indices 1, 2, 3, and Greek letters represent spacetime indices 0, 1, 2, 3.

What defines the spacetime 4-vector?

xµ = (x0, x1, x2, x3) = (ct, x, y, z) = (ct, r).

What does Lorentz invariance of Minkowski metric imply?

Λµ νΛρ σηµρ = ηνσ

What does the principle of covariance state about physical laws?

The principle of covariance states that physical laws (i.e. physical equations) should be tensorial, that is, in Minkowski space, they transform correctly under Lorentz transformations.

What is the 4-velocity?

uµ ≡ dxµ/dτ = γ(c, v).

What is 4-acceleration?

αµ = duµ/dτ = γ^2(0, a) + γ^4/c^2 * [v · a] (c, v)

Relativistically, what is 4-acceleration easiest to visualize?

In the comoving frame, where αµ = (0, a), for a the 3-acceleration in the comoving frame, in which the particle is instantaneously at rest (3-velocity zero).

When are 4-velocity and 4-acceleration orthogonal in spacetime?

In a comoving frame that is purely in the time and space directions.

What is 4-momentum?

P µ ≡ muµ = (E/c, P )

What best describes rest mass?

The value of the mass in the comoving frame Σ, denoted m.

Why is it important that dimensional analysis is useful in these cases?

Expressions being added, equated, or occurring in different components of a vector must have the same physical dimension. The fundamental physical dimensions for us are [ Length ], [ Time ], [ Mass ], and [ Charge ], ‘abbreviated’ [L], [T], [M], [Q]

What does the Minkowski equation show?

A 4-force cannot alter the magnitude of either velocity or momentum, but can only change the direction of the particle’s worldline.

For a charge q, what is Lorentz force?

F = q(E + v × B) acting on a charged particle.

In terms of 4-velocity and comoving charge density, what is the 4-current?

Jµ(x) = ρ0(x)uµ(x) = (cρ(x), J(x)).

Historically, what did scientists consider the ether as?

A material substance, and effects we now think of as relativistic were ascribed to material electromagnetic properties involving more complicated behaviour of D and H with respect to E and B.

Under Maxwell's equations in integral form, what does S represent?

Some 2D Surface in space, with the boundary as the closed loop ∂S.

In relation to finding the electromagnetic field tensor, what happens to space time?

We generalise the index notation for Euclidean 3D space to 3+1-dimensional Minkowski spacetime. Then, we adopt Einstein
s convention that roman letters i, k, . . . represent spatial indices 1,2,3, and greek letters µ, ν, ρ, σ, α, β, . . . represent spacetime indices 0,1,2,3.

What is the first step to understanding more formally Maxwell Equations in a tensor format?

In free space, Maxwell
s equations are written as:
∇ · B = 0,
∇ × E + B˙ = 0,
ε0∇ · E = ρ,
µ −1 ∇ × B − ε0E˙ = J.

What 4-vector component must transform in order to find a more fundamental understanding of momentum and energy as it relates to electromagnetism?

Spacetime coordinates, as for any 4-vector, given in the Σ frame as the component list xµ = (x0, x1, x2, x3) = (ct, x, y, z) = (ct, r).

Why do we define the Lorentz transformations to be those transformations which keep the Minkowski metric invariant?

Because it ensures ηµν has the same components in every spacetime coordinate system: Λµ νΛρ σηµρ = ηνσ.

The spacetime derivative is, ∂µ = (∂0, ∂1, ∂2, ∂3) = (c −1∂t, ∂x, ∂y, ∂z) = (c −1∂t, ∇). Therefore this transforms ∂µ = Λν µ∂ν, and what result is achieved?

The spacetime derivative of the spacetime vector is the same in every frame:
∂µxµ = ∂tt + ∂xx + ∂yy + ∂zz = 4.

How can more general definitions of terms in Special Relativity be expressed?

We define a contravariant 4-vector to be an array of four quantities vµ , µ = 0, 1, 2, 3 whose components, under Lorentz transformations, transform by the rule: vν = Λν µvµ , ν = 0, 1, 2, 3
A covariant 4-vector uµ is an array of four quantities (µ = 0, 1, 2, 3) which under Lorentz transforma- tions follow the rule : uν = Λµ νuµ, ν = 0, 1, 2, 3.

What is the Lorentz force using index format for Euclidean force?

F = q(E + v × B), or, more simply F = q E, or more expounded, Fi = q(Ei + εijkvjBk).

In what way did Maxwel make significant changes in physics?

Reconciled electricity, magnetism, and light.

Name a few of the major components of the Faraday Tensor

Exy, Exz, Ey, Ez, Bx, By, Bz

What does the metric tensor allow us to define?

Covariant and contravariant counterparts, vµ ≡ ηµνvν , for a contravariant 4-vector vν, and uµ ≡ ηµνuν for a covariant 4-vector uν.

In Minkowski Space, what describes length and time?

For spacetime tensors in Minkowski space, components of vectors being covariant and contravariant is simply a matter of having minus signs.

What is the spacetime interval?

s2 = ηµνxµxν = ηµνxµxν = xµxµ = (ct)2 − r · r = (ct)2 − r2