Kinematics — I

Revising my basics for an upcoming exam. Here are some of my notes on Relativistic Kinematics (Which can be used in Collider Physics too!)

PS no special prerequisite is required for the following.

I. Beta and Gamma : β and γ

Relativity tells us that the measurements of time, length, and other physical properties change for an object moving at the speeds comparable to time (Hence the term ‘Relative’). Special theory of relativity tells us :

1. The laws of physics take the same form in all inertial frames of reference.
2. As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body. Or: the speed of light in free space has the same value c in all inertial frames of reference.

Therefore, we must respect the speed of light and change other factors around it. That’s STR.

We continue our discussions by defining two important velocity dependent variables, β and γ (Lorentz Factor) :

The Lorentz Factor, gamma, along with the ratio of velocities for the basis of all the terms that we use in the relativistic treatment of physics. For Example :

Lorentz Transformations

A useful quantity to remember! -

γ and KE

Lorentz factor γ as a function of velocity. Its initial value is 1 (when v = 0); and as velocity approaches the speed of light (v → c) γ increases without bound (γ → ∞)

REMARK : value of γ is always >1. This will play a role in remembering things like length contraction or time evolution, etc (And you will avoid mistakes in your calculations)

We can divide the β vs γ plot into 3 regions based on the speed, v

Another factor that we might be interested in is the alpha, Lorentz inverse, the alpha-beta graph is given by :

Lorentz factor inverse

Expansion of γ in a Binomial Series :

Leading order, NLO, NNLO contributions to γ

I. Energy and Momentum

The word mass has two meanings in special relativity: invariant mass (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity of the observer. According to the concept of mass–energy equivalence, invariant mass is equivalent to rest energy, while relativistic mass is equivalent to relativistic energy (also called total energy).

rest and relativistic energies

Momentum, on the other hand, is something which can’t be talked about in one post, but here are some basics that you should know :

In special relativity, four-momentum (yes, revise your tensors) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is a four-vector in spacetime.

4 Momentum

Some basic properties of 4-momentum are as follows :

A Paradox?

In relativity it is very rare indeed to have any static potentials, such as a gravitational potential or an electrostatic potential to worry about. The reason is that potentials are not actually consistent with Einstein’s postulates. To see why this is, recall that the gravitational force between two bodies has magnitude GM1M2/r² where G is Newton’s gravitational constant, M1 and
M2 are the masses of the bodies and r is their separation. Now imagine that r is, say 30 kpc, roughly the dimension of the luminous component of our galaxy. Now let one of the bodies disappear suddenly. If Newton’s law of gravitation is correct, the force on the other mass disappears immediately! This is inconsistent with Einstein’s basic idea that nothing can travel faster than c, since if Newton’s law of gravitation is right, then the gravitational field can transmit information from place to place instantaneously. So, in relativity, static fields are replaced by propagating particles which carry the forces from place to place, never having a measured velocity faster than c.

This theory that a field is composed of force carrying particles gave rise to a new field, Quantum Field Theory, where the brilliant minds such as Dirac quantize the field. Voila! QFT!

Furthermore, because there are no potentials, there are no conventional forces either. Instead, bodies propagate from place to place in straight
lines until they undergo interactions, which we may consider to be approximately point scatters.

Scattering plays a HUGE role in Particle Physics, from reconstruction to the studies of jets, everything revolved around scattering and confinement principles of QFT.

A question persists : A great proportion of problems in special relativity has to do with studying the dynamics of collisions and decays. If you know the initial state of a system — say you have some particle at rest with respect to an observer O, and then something happens, say the particle decays into two bodies, what can you say about the final state of the system, from the perspective of observer O?

Conservation :

NOTE! Energy is only conserved if it is the same observer O measuring the energy of all the particles before the dynamic event (collision or decay), and measuring the energies of all the final state particles after the event. This is true for momentum conservation as well.

The sum of all the energies before the event is the same as the sum of all the energies after the event, as long as the same observer O measures the energies in both cases.

The sum of the momentum vectors of all the particles before and after the collision will be the same, as long as all those momentum vectors are determined by the same observer, non-accelerating throughout.