Modern Physics Unit 13: Special Relativity - Kinematics Lecture 13.3: Experimental Tests

Ron Reifenberger Professor of Physics Purdue University

1 How to verify time dilation? Requirements: • Object should move close to speed of light • Object must have an internal clock

Background - Lifetime of elementary particles • Many elementary particles are unstable.

• After some time has passed, they decay into other elementary particles.

• The average time until decay is called the particle's life-time.

• IDEA: Produce unstable particles, some at rest while others move around a particle accelerator with a speed close to c.

• When the lifetimes of the particles are measured, do the lifetime of unstable particles at rest have a significantly shorter lifetime than those in motion?

2 Muons fit the bill

The muon is an unstable, elementary particle similar to the electron:

. negative electric charge . spin of ½. . muon mass is about 200 greater than mass of electron.

Muons were discovered by Carl D. Anderson in 1936 while studying cosmic radiation.

Muons are by-products of cosmic rays colliding with molecules in the upper atmosphere. Muons reach earth with an average velocity of about 0.994c.

On earth’s surface, about 1 muon passes through a 1 cm2 area per minute (~10,000 muons per square meter in one minute). Muon flux is constant over Credits: CERN 3 time. What we know

The facts: vc= 0.994 As measured from a rest frame, Muons decay over time. They are unstable. The number β = 0.994 remaining as a function of time is given by: γ = t 9.1 τ 1 12 τ = × −6 Nt( )= Nt ( = 0)  Eq. (1) 12 1.5 10 s 2

Collection of muons at rest 1.5 µS

4 Writing it another way 1 Since= e−0.693 2 t τ12 t 1 −0.693 τ Nt( )= Nt ( = 0) =Nt( = 0)  e 12 2

0.693t − − t τ12 ( τ ) = Neoo= Ne Eq. (2) 1 0.693 where ≡ τ τ12

If ττ12 = 1.5µs, then = 2.16 µs Distinguish between half-life and lifetime

Different books use either of these two notations. Which time do you choose? It depends on whether you use Eq. (1) or Eq. (2).

In what follows, I use Eq. (1) because it is more intuitive.

5 The 1941 Mt. Washington, NH Test of Special Relativity

2000 m

6 Muon decay (Rossi and Hall, 1941) Muons created by cosmic With v= 0.994c, how far will a muon travel rays in upper atmosphere in a time interval Δt= τ½ ? distance= velocity × time ∆t 86− τ = ×× × × 1 12 0.994( 3 10ms /) ( 1.5 10 s) N(∆= t ) Nt ( = 0) 2 = 447 m Mt. Washington, NH Every 447 m of travel, the number of (2000 m ) muons will decrease by ½. h prediction 570/hr

t=0, h =2000 m; N(t=0)=570/hr (determined by experiment) h = 894 m Δ

Δt=2τ 12 , h=1106 m; N=142/hr h = 894 m

Δt=4 τ 12 , h=212 m; N=36/hr Δ

Δt=4.5 τ 12 , h=0 m; N=25/hr 7 What is measured?

Think of the 570/hr ∆=t 4.5τ mountain as 12 ∆t a 2000 m τ 1 12 long rod. N(∆= t ) Nt ( = 0) 2 4.5 1 = 570 / hr  = 25 / hr 2

BUT measurements give about 400/hr instead of 25/hr? Hmmm…. off by about a factor of about 16? What could go wrong? Are the Reference Frames consistent? one click

8 B. Sorting it Out A. Vf=0.994c; γ=9.1 Δt ’ = 1.5µs = τ’1/2

Vf=0.994c; γ=9.1 y Muon y’ S S’ 2000m ∆xxx =γ ∆'' ⇒∆ = 9.1 ∆=ttγ ∆' = 9.1 × 1.5µs = 220m x =13.6µτs = 12 x’ ∆x ' T ' = transit time =

2000 m ∆x T = transit time = 0.994c 0.994c 220m = x = x 2000m 8 ∆ = (0.994)⋅×( 3 10m / s) (0.994)⋅× 3 108 ms / ( ) = 0.74µs = 6.71µs T ' 0.74µs = = 0.49 Ts6.71µ τ µ = = 0.49 '12 1.5 s τ12 13.6µs

T τ 0.49 1112  one click N(')∆= t N =570 = 406 / hr o 22  9 Proper time Summary Proper length

Muon Earth’s Incorrect frame frame Muons at summit 570/hr 570/hr 570/hr Muon speed/ 0.994 c 0.994 c 0.994 c Earth speed (up) (down) Height of mountain 220 m 2000 m 2000 m T=Transit time 0.74 μs 6.71 μs 6.71 μs

Half-life (τ1/2) 1.5 μs 13.6 μs 1.5 μs

T/τ1/2 0.49 0.49 4.2 Two Number Surviving ~406/hr ~406/hr ~25/hr different frames! 1 Relativistic Parameters: ∆=tt' ∆ 0.74µµss×= 9.1 6.71 vc= 0.994 γ β = 0.994 γ = 9.1 1 ∆=xx ∆' 2000mm / 9.1= 220 γ 10 The common sense of relativity Relativity requires a deep appreciation for the measurement process and the speed at which information can be conveyed.

Relativity requires a new vocabulary and a new way of thinking

Space and Time do not exist separately, they are part of the same thing called Spacetime. Effectively, when you increase your speed through space, you decrease your “speed” through time.

All transformations and results are reciprocal; there is no experiment that you can perform to decide if you are at rest or moving at constant velocity

Word problems are especially tricky; trust the equations

Does time really slow down; do lengths really contract?

Time dilation + Length contraction + Lack of simultaneity = consistent framework 11

Up Next – relativistic velocity transformations