Lecture 5: Vectors in Spacetime and Lorentz Transformations in PHY312, Study notes of Physics

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PHY312 - lecture 5^ Simon Catterall

Review

Derived Lorentz transformations relating coordinates ofevents as seen by different inertial observers. Saw how to add velocities in special relativity. Examples Today: talk about

vectors in spacetime

Energy-momentum vector ...

Transformation between coordinates^ Taking dot products of first eqn. with respect to

′^ ileads

to

′^ ′ x=^ x(i.i) +

′ y(j.i)

′^ But i.i= cos (

′^ θ) and j.i= sin (

θ)^ so that the

transformation of the coordinates may be written

′^ x=^ cos (θ )x^ + sin (θ)y ′^ y=^ −^ sin (

θ)x^ + cos (θ)

y

Notice that the length of the position vector^2 2 x+^ y= (x

′^2 ′^2 )+ (y)is

invariant^ with respect to

transformation between different (rotated) coordinatesystems (although the component values

x^ and^ y^ are

certainly not).

Definition of vectors Can define^ 2d (space) vector as set of 2 numbers

(x, y)

which transform according to this rule as coordinatesystem is rotated. This rule is^

determined

by requiring that the length

2 2 x+^ ysame in all frames ... Sound familiar? Natural way to map the idea of an invariant spacetimeinterval under LT into the language of 2d rotations ...

Spacetime vectors

Define spacetime vector (sometimes called 4-vector) asa vector that transforms

same way as the spacetime

coordinate vector

Has 3 space and 1 time component.

(a, a, a, atxy

).z

2 Length a− t^

(^222) a−^ a−^ a x^ y^ z

same in all inertial FOR.

Components transform using LT. If theory formulated in terms of such vectors we areguaranteed that equations will look same in all inertialFOR! Can construct things everyone agrees on ..

Energy-moemntum vector Consider worldline of particle in spacetime. Any small portion of it looks like straight line withcomponents (in some FOR)

(c∆t,^ ∆x,^ ∆

y,^ ∆z)

This is simplest example of spacetime vector. Points along worldline at that point. Calculate proper time for this small displacement

∆τ^.

Consider vector

(c∆ P = m 0 t∆x∆y∆,^ ,^ ,^ ∆τ^ ∆τ^ ∆τ

)z ∆τ

E^ =^ mc

Consider small

v. Taylor expand the square rootdt mc =^ mc^00 dτ^

(^21) v(1 + +^... (^2 2) c

Contains the Newtonian kinetic energy

K/c^ plus

constant. It is a relativistic generalization of energy of motion. But notice it has a value even when at rest

2 mc! Rest 0

energy. But since

c^ is a constant shows that this rest

energy is just a measure of mass. Famous equivalence of mass and energy

2 E = mc. 0

Large amount of energy from small mass ...

Summary

Rederived LT by analogy with 2d spatial rotations.Concept of spacetime vector. Invariance of interval frominvariance of length. Laws of physics should be written in terms of suchvectors. Ensures principle of relativity ... Simplest example of such a vector – energy-momentumvector. Gives relativistic generalization of energy andmomentum. Shows equivalence of mass and energy.