Vectors

Four-vectors and conservation laws in relativity

This lecture was delivered to the final year honors class of 3 year science degree students on 21 November 2017 as part of the Classical Dynamics paper.

In this lecture we will discuss some of the important tools of relativistic mechanics. We will discuss the idea of proper-time, 4-velocity, 4-acceleration, 4-momentum, 4-force and related conservation law of the 4-momentum.

A. Proper-time.
The proper time is the time interval in the rest-frame of any event. The proper time is related to time-interval in other inertial frame by: tau = (1/gamma)t where gamma  > 1 always.

Gamma is the Lorentz factor or Lorentz boost factor directly related to the speed of an object in speed-of-light units, i.e. beta.

gamma = 1/sqrt{1-v^2/c^2}

Hence proper-time is the smallest possible time interval for an object in motion in among all possible inertial frames of reference and it occurs in the rest frame.

d(tau) < dt

Proper-time is necessary to define other basic quantities in theory of relativity if we are to preserve their basic meaning in terms of the non-relativistic mechanics definitions.

B. Four velocity.
Four velocity of a particle is the rate of change of 4-displacement …

So, …  is the position vector — or space-time interval in the Minkowski  space — akin to the difference of two 3-dimensional vector in coordinate space, this time with 4 coordinates rather than 3.

The proper-time interval d(tau) is a Lorentz invariant i.e. when we move between arbitrary inertial frames of references given by the Lorentz factor beta or  gamma this interval retains its value — because it retains its form. Any variable which would retain its form under such transformation are said to be Lorentz invariant quantities.

Relativistic Doppler effect

Relativistic Doppler effect.

There is an apparent shift in the observed frequency of any electromagnetic wave (light) when there is any relative motion between the source of light and the observer. This can be easily determined by using the 4-vector formulation of theory of relativity.

Lets discuss the details of this phenomena under two situations.

A. Source is at rest and observer is in motion.
Lets us consider two inertial frames S and S’. S’ is moving wrt S, along the x-axis with speed v = (beta) c where the observer is at rest in S’ frame but the source is at rest in the  S frame.

Wrong question in GATE 2018 physics?

I think the above question asked in GATE 2018 (physics) is wrong.

Any vector has two components. The component perpendicular to the parity axis has even parity and the parallel component to the axis has odd parity.

The opposite is true for axial vectors.

E, A vectors.
B, L axial vectors.

The correct answer per gate exam body is E, A. Why not B and L? It’s an arbitrary situation and perpendicular components of these fields will have odd parity.

Cross and Dot product of vectors.

Someone asked a very interesting question on the role of vectors in Physics. He was curious to know if dot product of vectors is natural but vector product is just syncretism, — that is make shift or unnatural manipulation.

Every vector can be resolved into two components. The cosine and sine components (any two vectors would constitute a plane) while cos part can represent the projection defined through dot PDT we can’t leave out the sine part. It plays its role through the vector or cross PDT.

The vector direction is no more along same direction as original vectors because of orthogonality. To preserve symmetry of both orthogonal components (or equal footing of both vectors, vector a and b eg) we need the 3rd dimension. Hence such a definition of cross PDT.

Eg the emf generated in a changing “mag field” (Faraday’s law) depends on change in mag field if area is held const. It also depends on change in “area vector” if mag field is held const. So there are two vectors involved and their transverse values matter (and not their longitudinal values). To preserve equivalent role of both area and mag field vectors the resultant vector must be in a 3rd orthogonal direction …

Also think of this; a scalar is not necessarily directionless. (think electric current or even temperature or heat gradient etc) They just do not have the full fledged capacity of vectors. Its like flower bud vs fully blossomed flower.

So scalars can’t be added like vectors. We tend to make a mistake here. We say scalars don’t have a direction. That’s totally erroneous. They do have direction and it matters. Which direction you want to stick to if the current flows along certain direction only?

Lets make it still more clear.

If there are two directions in which there are electric currents, we say they are both equal, the direction won’t matter. That’s where we make the mistake. We should say they are equivalent and not equal. Equality is ideal, its mathematical. But equivalence is physical. Its the effects of both currents in a certain sense that make them equivalent, but their strict equality does not follow.

Interference of two plane harmonic waves

Optics Series Lecture, Lecture – VII.

“Conditions of interference, Interference of two plane harmonic waves.”

This lecture was delivered on 7th February in a lecture session of 1 and 1/2 hours. This lecture was delivered to Physics elective students but intended as a lecture towards Honors students at a later date.

Electromagnetic Waves.
Light is an electromagnetic wave. In-fact its a transverse electromagnetic wave which means the oscillation of E and B fields produces light which propagates in a direction that is perpendicular to the plane that contains the E and B fields. In other words E, B and k the vector that denotes the direction of light propagation, are mutually perpendicular vectors.

We will study these details in a later intended lecture. EM waves are not only transverse waves but also vector waves, that is; E and B are vector fields whose undulation is summarized as light.

Light is a general name for all EM waves but visible light is that particular part of EM waves which has frequency of wave such that the wavelength varies from approximately 400 – 700 nm. In vacuum — only in vacuum, light always moves at a fixed speed: namely c = 3×108 m/s. Therefore light whose wavelength lies between 400 – 700 nm is called as visible light: we can write in vacuum c = νλ.

Light as a transverse wave phenomenon of vector fields is comprehensively described by four equations known as Maxwell’s Equations. The Maxwell’s Equations are a summary of important and fundamental laws of electricity and magnetism — together called as electromagnetism, such as Gauss Law and Ampere’s Law. These equations produce the wave equation of motion, a linear, homogeneous, 2nd order differential equation that we will study a few lectures afterwards.

If you are quite serious and technically well equipped though, you can have a glimpse of it all — and may be work out to your satisfaction, by following the link to my slide-share presentations. There are many other important Physics concepts that are worked out in great detail, in those slide-share presentations by me. eg check: Electromagnetic Waves.

Let us therefore write the wave equation of motion, where the 3 components of E field — such as Ex, Ey or Ez or the 3 components of B field such as Bx, By, Bz, are denoted as ψ chosen anyone at one time. eg we can chose Ex = ψ. In general we have: