The Mathematics of Special Relativitys-jruiz8/SeniorResearch/presentation.pdf · Essentials of...

Essentials of RelativityThe Space of Relativity

The Mathematics of Special Relativity

Jared RuizAdvised by Dr. Steven Kent

May 6, 2009

Jared RuizAdvised by Dr. Steven Kent The Mathematics of Special Relativity

Table of Contents

Essentials of Relativity

HistoryImportant DefinitionsTime DilationLorentz Transformation

Space of Relativity

IntervalPredictionsTopology

History

1632: Galileo Galilei publishes Dialogue concerning the twochief world systems

1687: Isaac Newton releases Philosophiae Naturalis PrincipiaMathematica.

1865: James Clerk Maxwell unifies the theories of electricityand magnetism.

∇ · D = 4πρ

∇ · B = 0

∇× H =4π

∇× E +1

∂t= 0

History

∇ · D = 4πρ

∇ · B = 0

∇× H =4π

∇× E +1

∂t= 0

History

∇ · D = 4πρ

∇ · B = 0

∇× H =4π

∇× E +1

∂t= 0

History

∇ · D = 4πρ

∇ · B = 0

∇× H =4π

∇× E +1

∂t= 0

History

∇ · D = 4πρ

∇ · B = 0

∇× H =4π

∇× E +1

∂t= 0

1887: Null result of the Michelson-Morley Experiment.

1905: Albert Einstein reveals the Theory of Special Relativity.

Postulate 1

There is no absolute standard of rest; only relative motion isobservable.

Postulate 2

The velocity of light c is independent of the motion of the source.

Postulate 1

Postulate 2

Postulate 1

Postulate 2

Postulate 1

Postulate 2

Definitions

Definition

The four-dimensional set of axes is known as spacetime.

Definition

In classical mechanics, motion is described in a frame of reference.

Definitions

Definition

Definitions

Definition

A reference frame is inertial if every test particle initially at restremains at rest and every particle in motion remains in motionwithout changing speed or direction.

Definition

An event is a point (t, x , y , z) in spacetime.

Definition

Any line which joins different events associated with a given objectwill be called a world-line.

Definition

An observer is a series of clocks in an inertial reference framewhich measures the time.

Definition

Theorem

Given two observers, O and O ′, events in the inertial frames ofreference set up by the observers are related by:

t ′ = t

x ′ = x + vt

y ′ = y

z ′ = z

where v is the relative velocity which O ′ is moving with respect toO.

This theorem is incorrect (physically). We have to reexamine ourideas of time.

Theorem

Given two observers, O and O ′, events in the inertial frames ofreference set up by the observers are related by:

t ′ = t

x ′ = x + vt

y ′ = y

z ′ = z

where v is the relative velocity which O ′ is moving with respect toO.

This theorem is incorrect (physically). We have to reexamine ourideas of time.

Definition

The proper time τ along the worldline of a particle in constantmotion is the time measured in an inertial coordinate system inwhich the particle is at rest.

If we assume Postulate 2 by Einstein (that c is a constant), thent = kt ′. But if O ′ is at rest relative to O, t ′ = kt. This k is knownas Bondi’s k-factor.

Definition

If we assume Postulate 2 by Einstein (that c is a constant), thent = kt ′.

But if O ′ is at rest relative to O, t ′ = kt. This k is knownas Bondi’s k-factor.

Definition

If we assume Postulate 2 by Einstein (that c is a constant), thent = kt ′. But if O ′ is at rest relative to O, t ′ = kt.

This k is knownas Bondi’s k-factor.

Definition

If we assume Postulate 2 by Einstein (that c is a constant), thent = kt ′. But if O ′ is at rest relative to O, t ′ = kt. This k is knownas Bondi’s k-factor.

Bondi’s k-factor

According to O, t = kt ′, where k is a constant. According toO ′, t ′ = kt.tB and dB can be expressed as:

2c(k2 − 1)t and tB =

2(k2 + 1)t.

Bondi’s k-factor

2(k2 + 1)t.

Bondi’s k-factor

2(k2 + 1)t.

Bondi’s k-factor

2c(k2 − 1)t

and tB =1

2(k2 + 1)t.

Bondi’s k-factor

2(k2 + 1)t.

The Gamma Factor

12c(k2 − 1)t12(k2 + 1)t

=c(k2 − 1)

(k2 + 1).

vk2 + v = ck2 − c

c + v = ck2 − vk2

c + v = k2(c − v)

√c + v

c − v> 1.

time E to B measured by O

time E to B measured by O ′=

=(k2 + 1)t

2kt= γ(v)

γ(v) =(k2 + 1)t

1√1− v2/c2.

The Gamma Factor

12c(k2 − 1)t12(k2 + 1)t

=c(k2 − 1)

(k2 + 1).

vk2 + v = ck2 − c

c + v = ck2 − vk2

c + v = k2(c − v)

√c + v

c − v> 1.

=(k2 + 1)t

2kt= γ(v)

γ(v) =(k2 + 1)t

1√1− v2/c2.

The Gamma Factor

12c(k2 − 1)t12(k2 + 1)t

=c(k2 − 1)

(k2 + 1).

vk2 + v = ck2 − c

c + v = ck2 − vk2

c + v = k2(c − v)

√c + v

c − v> 1.

=(k2 + 1)t

2kt= γ(v)

γ(v) =(k2 + 1)t

1√1− v2/c2.

The Gamma Factor

12c(k2 − 1)t12(k2 + 1)t

=c(k2 − 1)

(k2 + 1).

vk2 + v = ck2 − c

c + v = ck2 − vk2

c + v = k2(c − v)

√c + v

c − v> 1.

=(k2 + 1)t

2kt= γ(v)

γ(v) =(k2 + 1)t

1√1− v2/c2.

The Lorentz Transformation

Theorem

The inertial coordinate systems set up by two observers are relatedby:

t =t ′ + (vx ′/c2)√

1− (v/c)2

x =x ′ + vt ′√1− (v/c)2

We can write this more concisely as(ctx

)= γ(v)

(1 v/c

)(ct ′

where v is the relative velocity.

The Lorentz Transformation

Theorem

The inertial coordinate systems set up by two observers are relatedby:

t =t ′ + (vx ′/c2)√

1− (v/c)2

x =x ′ + vt ′√1− (v/c)2

We can write this more concisely as(ctx

)= γ(v)

(1 v/c

)(ct ′

where v is the relative velocity.

First consider two observers moving at constant speeds.We express an event in terms of both coordinate systems.We substitute in for Bondi’s k-factor(

(k + k−1 k − k−1

k − k−1 k + k−1

)(ct ′

)After some algebra and simplification, we have our result 2.

First consider two observers moving at constant speeds.

We express an event in terms of both coordinate systems.We substitute in for Bondi’s k-factor(

(k + k−1 k − k−1

k − k−1 k + k−1

)(ct ′

First consider two observers moving at constant speeds.We express an event in terms of both coordinate systems.

We substitute in for Bondi’s k-factor(ctx

(k + k−1 k − k−1

k − k−1 k + k−1

)(ct ′

(k + k−1 k − k−1

k − k−1 k + k−1

)(ct ′

(k + k−1 k − k−1

k − k−1 k + k−1

)(ct ′

Velocity Boost

In four dimensions,ctxyz

γ γv/c 0 0

γv/c γ 0 00 0 1 00 0 0 1

ct ′

where v is the relative velocity and γ = γ(v).

Definition

The 4× 4 matrix in (2) is known as the boost, denoted Lv .

With the properties of the boost, we can look at how an observerwould judge a moving particle’s velocity.

Velocity Boost

In four dimensions,ctxyz

γ γv/c 0 0

γv/c γ 0 00 0 1 00 0 0 1

ct ′

where v is the relative velocity and γ = γ(v).

Definition

The 4× 4 matrix in (2) is known as the boost, denoted Lv .

With the properties of the boost, we can look at how an observerwould judge a moving particle’s velocity.

Theorem

A particle cannot travel faster than c , the velocity of light.

Proof: (ct ′

)= γ(u)

(1 −u/c−u/c 1

−vt + b

w = −dx ′

dt ′=

1 + uv/c2.

Theorem

Proof: (ct ′

)= γ(u)

(1 −u/c−u/c 1

−vt + b

w = −dx ′

dt ′=

1 + uv/c2.

Theorem

Proof: (ct ′

)= γ(u)

(1 −u/c−u/c 1

−vt + b

w = −dx ′

dt ′=

1 + uv/c2.

w =v + u

1 + uv/c2.

Letting |u| < c and |v | < c , we see that |w | < c since

(c − u)(c − v) > 0 ⇐⇒ −(u + v)c > −c2 − uv

u + v < c(

)w < c .

(c + u)(c + v) > 0 ⇐⇒ (u + v)c > −c2 − uv

u + v > −c(

)w > −c . 2

w =v + u

1 + uv/c2.

(c − u)(c − v) > 0 ⇐⇒ −(u + v)c > −c2 − uv

u + v < c(

)w < c .

(c + u)(c + v) > 0 ⇐⇒ (u + v)c > −c2 − uv

u + v > −c(

)w > −c . 2

w =v + u

1 + uv/c2.

(c − u)(c − v) > 0 ⇐⇒ −(u + v)c > −c2 − uv

u + v < c(

)w < c .

(c + u)(c + v) > 0 ⇐⇒ (u + v)c > −c2 − uv

u + v > −c(

)w > −c . 2

In four-dimensions, we can say that

ct ′

where L =

(1 00 H

(1 00 KT

)and H,K are 3× 3

orthogonal matrices.

Definition

The matrix L in (3) is a Lorentz transformation if L−1 = gLT g ,

where g =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

. L is orthochronous if l1,1 > 0,

where l1,1 is the first entry of the first row of L

In four-dimensions, we can say thatctxyz

ct ′

where L =

(1 00 H

(1 00 KT

)and H,K are 3× 3

Definition

where g =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

ct ′

where L =

(1 00 H

(1 00 KT

)and H,K are 3× 3

Definition

The matrix L in (3) is a Lorentz transformation if

L−1 = gLT g ,

where g =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

ct ′

where L =

(1 00 H

(1 00 KT

)and H,K are 3× 3

Definition

where g =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

L is orthochronous if l1,1 > 0,

ct ′

where L =

(1 00 H

(1 00 KT

)and H,K are 3× 3

Definition

where g =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

Definition

In special relativity, the space we live in is called Minkowski Space,denoted by M = R× R3 where R = R× 0 is the time axes, andR3 = 0× R3 the space axes.

The distance D from a point to the origin is D =√

x2 + y2 + z2.

Emit a light pulse when t = x = y = z = 0.

It arrives at (ct, x , y , z) if ct =√

x2 + y2 + z2 = D.

c2t2 = D2, and c2t2 − x2 − y2 − z2 = 0.

Definition

In Minkowski space, the interval between any two eventsx = (t1, x1, y1, z1) and y = (t2, x2, y2, z2) is defined to be

c2(t2 − t1)2 − (x2 − x1)2 − (y2 − y1)2 − (z2 − z1)2.

Definition

x2 + y2 + z2.

x2 + y2 + z2 = D.

c2t2 = D2, and c2t2 − x2 − y2 − z2 = 0.

Definition

c2(t2 − t1)2 − (x2 − x1)2 − (y2 − y1)2 − (z2 − z1)2.

Definition

x2 + y2 + z2.

x2 + y2 + z2 = D.

c2t2 = D2, and c2t2 − x2 − y2 − z2 = 0.

Definition

c2(t2 − t1)2 − (x2 − x1)2 − (y2 − y1)2 − (z2 − z1)2.

Definition

x2 + y2 + z2.

x2 + y2 + z2 = D.

c2t2 = D2, and c2t2 − x2 − y2 − z2 = 0.

Definition

c2(t2 − t1)2 − (x2 − x1)2 − (y2 − y1)2 − (z2 − z1)2.

Definition

x2 + y2 + z2.

x2 + y2 + z2 = D.

c2t2 = D2, and c2t2 − x2 − y2 − z2 = 0.

Definition

c2(t2 − t1)2 − (x2 − x1)2 − (y2 − y1)2 − (z2 − z1)2.

Definition

x2 + y2 + z2.

x2 + y2 + z2 = D.

c2t2 = D2, and c2t2 − x2 − y2 − z2 = 0.

Definition

c2(t2 − t1)2 − (x2 − x1)2 − (y2 − y1)2 − (z2 − z1)2.

Invariance of the Interval

c2(t2 − t1)2 − (x2 − x1)2 − (y2 − y1)2 − (z2 − z1)2 = 0

for an observer O,

c2(t ′2 − t ′1)2 − (x ′2 − x ′1)2 − (y ′2 − y ′1)2 − (z ′2 − z ′1)2 = 0

for an observer O ′, moving with constant velocity relative to O.Because of this, we say the interval is invariant.

c2(t2 − t1)2 − (x2 − x1)2 − (y2 − y1)2 − (z2 − z1)2 = 0

for an observer O, then

c2(t ′2 − t ′1)2 − (x ′2 − x ′1)2 − (y ′2 − y ′1)2 − (z ′2 − z ′1)2 = 0

for an observer O ′, moving with constant velocity relative to O.

Because of this, we say the interval is invariant.

c2(t2 − t1)2 − (x2 − x1)2 − (y2 − y1)2 − (z2 − z1)2 = 0

for an observer O, then

c2(t ′2 − t ′1)2 − (x ′2 − x ′1)2 − (y ′2 − y ′1)2 − (z ′2 − z ′1)2 = 0

for an observer O ′, moving with constant velocity relative to O.Because of this, we say the interval is invariant.

The Inner Product

Definition

In M, two objects X = (X0,X1,X2,X3) and X ′ = (X ′0,X′1,X

′2,X

are called four-vectors ifX = LX ′

where L is the general Lorentz transformation.

For x = (ct1, x1, y1, z1), y = (ct2, x2, y2, z2) ∈M, the displacementfour-vectorX = y − x = c(t2 − t1) + (x2 − x1) + (y2 − y1) + (z2 − z1).

Definition

The inner product between two four-vectors X ,Y ∈M is:

g(X ,Y ) = X0Y0 − X1Y1 − X2Y2 − X3Y3.

The Inner Product

Definition

′2,X

Definition

g(X ,Y ) = X0Y0 − X1Y1 − X2Y2 − X3Y3.

The Inner Product

Definition

′2,X

Definition

g(X ,Y ) = X0Y0 − X1Y1 − X2Y2 − X3Y3.

Predictions

Definition

The four-velocity of a particle, (V0,V1,V2,V3), is given by:

V0 = cdt

dτ, V1 =

dτ, V2 =

dτ, V3 =

Theorem

Let an observer O be moving with constant velocity V . Then Osees two events A and B as being simultaneous if and only if thedisplacement g(X ,V ) = 0, where X = B − A.

Corollary

Two events which are simultaneous for one observer may not besimultaneous for another observer moving at constant velocity withthe respect to the first observer.

Predictions

Definition

V0 = cdt

dτ, V1 =

dτ, V2 =

dτ, V3 =

Theorem

Corollary

Predictions

Definition

V0 = cdt

dτ, V1 =

dτ, V2 =

dτ, V3 =

Theorem

Corollary

The Lorentz Contraction

Theorem

If a rod has length R0 in a rest frame, then in an inertial coordinatesystem oriented in the direction of the unit vector e and movingwith respect to the rod with velocity v, the length of the rod is:

√c2 − v2√

c2 − v2 sin2(θ)

where θ is the angle between e and v.

The Lorentz Contraction

Theorem

If a rod has length R0 in a rest frame, then in an inertial coordinatesystem oriented in the direction of the unit vector e and movingwith respect to the rod with velocity v, the length of the rod is:

√c2 − v2√

c2 − v2 sin2(θ)

where θ is the angle between e and v.

Definition

The rest mass m0 = m(0) of a body is the mass of a bodymeasured in an inertial coordinate system in which the body is atrest.

Theorem

A body’s inertial mass m is a function of v , and can be rewrittenas m = m(v) = γ(v)m0.

Theorem

E = mc2.

Definition

Theorem

E = mc2.

Definition

Theorem

E = mc2.

Back to the Inner Product

Recall: g(X ,Y ) = X0Y0 − X1Y1 − X2Y2 − X3Y3.

For any three four-vectors X ,Y ,Z ∈M, and α ∈ R:

g(X ,Y ) = g(Y ,X ).