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

cJ +

1

c

∂D

∂t

∇× E +1

c

∂B

∂t= 0



History




∇ · D = 4πρ

∇ · B = 0

∇× H =4π

cJ +

1

c

∂D

∂t

∇× E +1

c

∂B

∂t= 0



History




∇ · D = 4πρ

∇ · B = 0

∇× H =4π

cJ +

1

c

∂D

∂t

∇× E +1

c

∂B

∂t= 0



History




∇ · D = 4πρ

∇ · B = 0

∇× H =4π

cJ +

1

c

∂D

∂t

∇× E +1

c

∂B

∂t= 0



History




∇ · D = 4πρ

∇ · B = 0

∇× H =4π

cJ +

1

c

∂D

∂t

∇× E +1

c

∂B

∂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


Definition




Definitions

Definition


Definition




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


Definition


Definition


Definition




Definition


Definition


Definition


Definition




Definition


Definition


Definition


Definition




Definition


Definition


Definition


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:

dB =1

2c(k2 − 1)t and tB =

1

2(k2 + 1)t.



Bondi’s k-factor


dB =1


1

2(k2 + 1)t.



Bondi’s k-factor


dB =1


1

2(k2 + 1)t.



Bondi’s k-factor


dB =1

2c(k2 − 1)t

and tB =1

2(k2 + 1)t.



Bondi’s k-factor


dB =1


1

2(k2 + 1)t.



The Gamma Factor

v =dB

tB=

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)

k =

√c + v

c − v> 1.

time E to B measured by O

time E to B measured by O ′=

tBkt

=(k2 + 1)t

2kt= γ(v)

γ(v) =(k2 + 1)t

2kt=

1√1− v2/c2.



The Gamma Factor

v =dB

tB=

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)

k =

√c + v

c − v> 1.



tBkt

=(k2 + 1)t

2kt= γ(v)

γ(v) =(k2 + 1)t

2kt=

1√1− v2/c2.



The Gamma Factor

v =dB

tB=

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)

k =

√c + v

c − v> 1.



tBkt

=(k2 + 1)t

2kt= γ(v)

γ(v) =(k2 + 1)t

2kt=

1√1− v2/c2.



The Gamma Factor

v =dB

tB=

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)

k =

√c + v

c − v> 1.



tBkt

=(k2 + 1)t

2kt= γ(v)

γ(v) =(k2 + 1)t

2kt=

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

v/c 1

)(ct ′

x ′

)(1)

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

v/c 1

)(ct ′

x ′

)(1)

where v is the relative velocity.



Proof

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

)=

1

2

(k + k−1 k − k−1

k − k−1 k + k−1

)(ct ′

x ′

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



Proof

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

)=

1

2

(k + k−1 k − k−1

k − k−1 k + k−1

)(ct ′

x ′




Proof

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

)=

1

2

(k + k−1 k − k−1

k − k−1 k + k−1

)(ct ′

x ′




Proof


ctx

)=

1

2

(k + k−1 k − k−1

k − k−1 k + k−1

)(ct ′

x ′




Proof


ctx

)=

1

2

(k + k−1 k − k−1

k − k−1 k + k−1

)(ct ′

x ′




Velocity Boost

In four dimensions,ctxyz

=

γ γv/c 0 0

γv/c γ 0 00 0 1 00 0 0 1

ct ′

x ′

y ′

z ′

(2)

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 ′

x ′

y ′

z ′

(2)

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 ′

x ′

)= γ(u)

(1 −u/c−u/c 1

)(ct

−vt + b

).

w = −dx ′

dt ′=

v + u

1 + uv/c2.



Theorem


Proof: (ct ′

x ′

)= γ(u)

(1 −u/c−u/c 1

)(ct

−vt + b

).

w = −dx ′

dt ′=

v + u

1 + uv/c2.



Theorem


Proof: (ct ′

x ′

)= γ(u)

(1 −u/c−u/c 1

)(ct

−vt + b

).

w = −dx ′

dt ′=

v + u

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(

1 +uv

c2

)w < c .

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

u + v > −c(

1 +uv

c2

)w > −c . 2



w =v + u

1 + uv/c2.


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

u + v < c(

1 +uv

c2

)w < c .

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

u + v > −c(

1 +uv

c2

)w > −c . 2



w =v + u

1 + uv/c2.


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

u + v < c(

1 +uv

c2

)w < c .

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

u + v > −c(

1 +uv

c2

)w > −c . 2



In four-dimensions, we can say that

ctxyz

= L

ct ′

x ′

y ′

z ′

(3)

where L =

(1 00 H

)Lv

(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

= L

ct ′

x ′

y ′

z ′

(3)

where L =

(1 00 H

)Lv

(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

ct ′

x ′

y ′

z ′

(3)

where L =

(1 00 H

)Lv

(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






= L

ct ′

x ′

y ′

z ′

(3)

where L =

(1 00 H

)Lv

(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,





= L

ct ′

x ′

y ′

z ′

(3)

where L =

(1 00 H

)Lv

(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

If

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.




If

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.




If

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

′3)

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

′3)




Definition


g(X ,Y ) = X0Y0 − X1Y1 − X2Y2 − X3Y3.



The Inner Product

Definition


′2,X

′3)




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 =

dx

dτ, V2 =

dy

dτ, V3 =

dz

dτ.

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 =

dx

dτ, V2 =

dy

dτ, V3 =

dz

dτ.

Theorem


Corollary




Predictions

Definition


V0 = cdt

dτ, V1 =

dx

dτ, V2 =

dy

dτ, V3 =

dz

dτ.

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:

R =R0

√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:

R =R0

√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


Theorem

E = mc2.



Definition


Theorem


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 ).

g(αX + βY ,Z ) = αg(X ,Z ) + βg(Y ,Z ).

But g(X ,Y ) < 0 if X0Y0 < X1Y1 + X2Y2 + X3Y3.

So g is an indefinite inner product. Thus g(X ,X ) (or the interval)can be used to split Minkowski space into cones.






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

g(αX + βY ,Z ) = αg(X ,Z ) + βg(Y ,Z ).








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

g(αX + βY ,Z ) = αg(X ,Z ) + βg(Y ,Z ).








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

g(αX + βY ,Z ) = αg(X ,Z ) + βg(Y ,Z ).








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

g(αX + βY ,Z ) = αg(X ,Z ) + βg(Y ,Z ).








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

g(αX + βY ,Z ) = αg(X ,Z ) + βg(Y ,Z ).








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

g(αX + βY ,Z ) = αg(X ,Z ) + βg(Y ,Z ).


So g is an indefinite inner product.

Thus g(X ,X ) (or the interval)can be used to split Minkowski space into cones.






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

g(αX + βY ,Z ) = αg(X ,Z ) + βg(Y ,Z ).





Definition

For an event x ∈M, we have:

Light-Cone CL(x) = {y : g(X ,X ) = 0}Time-Cone CT (x) = {y : g(X ,X ) > 0}Space-Cone CS(x) = {y : g(X ,X ) < 0}

where y ∈M, and X is the displacement vector from x to y



Definition






Definition


Light-Cone CL(x) = {y : g(X ,X ) = 0}

Time-Cone CT (x) = {y : g(X ,X ) > 0}Space-Cone CS(x) = {y : g(X ,X ) < 0}




Definition


Light-Cone CL(x) = {y : g(X ,X ) = 0}Time-Cone CT (x) = {y : g(X ,X ) > 0}

Space-Cone CS(x) = {y : g(X ,X ) < 0}where y ∈M, and X is the displacement vector from x to y



Definition






Definition

For events x , y ∈M we define a partial ordering < on M by x < yif the displacement vector X from x to y lies in the futurelight-cone

That is, x < y if ty > tx , and g(X ,X ) > 0.



Definition

For events x , y ∈M we define a partial ordering < on M by x < yif the displacement vector X from x to y lies in the futurelight-cone

That is, x < y if ty > tx , and g(X ,X ) > 0.



Definition


where g =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

.


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

Definition

Define the Lorentz groupL = {λ : M→M : ∀X ,Y ∈M, g(λX , λY ) = g(X ,Y )}

g(X ′,Y ′) = g(LX , LY ) = g(λX , λY ) = g(X ,Y ).

Definition

The orthochronous Lorentz group L+ is the subgroup of L whoseelements preserve the partial ordering < on M.



Definition


where g =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

.



Definition



Definition




Definition


where g =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

.



Definition


g(X ′,Y ′)

= g(LX , LY ) = g(λX , λY ) = g(X ,Y ).

Definition




Definition


where g =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

.



Definition


g(X ′,Y ′) = g(LX , LY )

= g(λX , λY ) = g(X ,Y ).

Definition




Definition


where g =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

.



Definition


g(X ′,Y ′) = g(LX , LY ) = g(λX , λY )

= g(X ,Y ).

Definition




Definition


where g =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

.



Definition



Definition




Definition


where g =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1



Definition



Definition




Definition


where g =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1



Definition



Definition




Usually we think of the topology on R4 as the standard Euclideantopology T .

Physically, this topology is not useful because:

1. The 4-dimensional Euclidean topology is locally homogeneous,yet M is not (the light cone separates timelike and spacelikeevents).

2. The group of all homeomorphisms of R4 include mappings ofno physical significance.

Two properties which T F will satisfy are:

1*. T F is not locally homogenous, and the light cone through anypoint can be deduced from T F .

2*. The group of all homeomorphisms of the fine topology isgenerated by the inhomogeneous Lorentz group anddilatations.



Usually we think of the topology on R4 as the standard Euclideantopology T .Physically, this topology is not useful because:
































Definition

Define the group G to be the group consisting of:

1 The Lorentz group L.

2 Translations (X ′ = X + K where K ∈M is constant).

3 Multiplication by a scalar, or dilatations (X ′ = αX for α ∈ R).

Definition

Define the group G0 to be the group consisting of:

1 The orthochronous Lorentz group L+.

2 Translations.

3 Dilatations.



Definition

Define the group G to be the group consisting of:

1 The Lorentz group L.

2 Translations (X ′ = X + K where K ∈M is constant).

3 Multiplication by a scalar, or dilatations (X ′ = αX for α ∈ R).

Definition

Define the group G0 to be the group consisting of:

1 The orthochronous Lorentz group L+.

2 Translations.

3 Dilatations.



Definition

The fine topology T F of M induces the 1-dimensional Euclideantopology on gR and the 3-dimensional topology on gR3, whereg ∈ G .

A set U ∈M is open in T F ⇐⇒ U ∩ gR is open in gR,and U ∩ gR3 is open in gR3.

The ε-neighborhoods in T are NEε (x) = {y : d(x , y) < ε}.

Definition

We define the ε-neighborhoods in T F as

NFε (x) = NE

ε (x) ∩(

CT (x) ∪ CS(x))

where x ∈M.



Definition

The fine topology T F of M induces the 1-dimensional Euclideantopology on gR and the 3-dimensional topology on gR3, whereg ∈ G . A set U ∈M is open in T F ⇐⇒ U ∩ gR is open in gR,and U ∩ gR3 is open in gR3.


Definition


NFε (x) = NE

ε (x) ∩(

CT (x) ∪ CS(x))

where x ∈M.



Definition



Definition


NFε (x) = NE

ε (x) ∩(

CT (x) ∪ CS(x))

where x ∈M.



Definition



Definition


NFε (x) = NE

ε (x) ∩(

CT (x) ∪ CS(x))

where x ∈M.



Theorem

NFε (x) is open in T F .

Proof: Let A be either the time axes or space axes. Then

NFε (x) ∩ A =

{NEε (x) ∩ A if x ∈ A(NEε (x) \ CL(x)

)∩ A if x /∈ A



Theorem



NFε (x) ∩ A =

{

NEε (x) ∩ A if x ∈ A(NEε (x) \ CL(x)

)∩ A if x /∈ A



Theorem



NFε (x) ∩ A =

{NEε (x) ∩ A if x ∈ A

(NEε (x) \ CL(x)

)∩ A if x /∈ A



Theorem



NFε (x) ∩ A =


)∩ A if x /∈ A



Theorem



NFε (x) ∩ A =


)∩ A if x /∈ A



Theorem



NFε (x) ∩ A =


)∩ A if x /∈ A



First Result

Theorem

The group of all homeomorphisms of T F is G .

Proof: T F is defined invariantly under G , so every g ∈ G is ahomeomorphism. Now we have to show every homeomorphismh :(M, T F

)→(M, T F

)is in G .

Now let g ∈ G be the element corresponding to time reflection.

Lemma

Let h :(M, T F

)→(M, T F

)be a homeomorphism. Then h either

preserves the partial ordering or reverses it.

So either h or gh preserves the partial ordering, and is an elementof G0.The group of all homeomorphisms is thus generated by g and G0,which is G . 2



First Result

Theorem



)→(M, T F

)is in G .


Lemma

Let h :(M, T F

)→(M, T F






First Result

Theorem


Proof: T F is defined invariantly under G , so every g ∈ G is ahomeomorphism.

Now we have to show every homeomorphismh :(M, T F

)→(M, T F

)is in G .


Lemma

Let h :(M, T F

)→(M, T F






First Result

Theorem



)→(M, T F

)is in G .


Lemma

Let h :(M, T F

)→(M, T F






First Result

Theorem



)→(M, T F

)is in G .


Lemma

Let h :(M, T F

)→(M, T F






First Result

Theorem



)→(M, T F

)is in G .


Lemma

Let h :(M, T F

)→(M, T F



So either h or gh preserves the partial ordering, and is an elementof G0.

The group of all homeomorphisms is thus generated by g and G0,which is G . 2



First Result

Theorem



)→(M, T F

)is in G .


Lemma

Let h :(M, T F

)→(M, T F






Second Result

Corollary

The light, time, and space cones through a point can be deducedfrom the topology.

Proof:For x ∈ M, let Gx be the group of homeomorphisms whichfix x . By the theorem, Gx is generated by the Lorentz group anddilatations. Therefore there are exactly four orbits under Gx :CL(x) \ x , CT (x) \ x , CS(x) \ x , the point x . 2



Second Result

Corollary


Proof:For x ∈ M, let Gx be the group of homeomorphisms whichfix x .

By the theorem, Gx is generated by the Lorentz group anddilatations. Therefore there are exactly four orbits under Gx :CL(x) \ x , CT (x) \ x , CS(x) \ x , the point x . 2



Second Result

Corollary


Proof:For x ∈ M, let Gx be the group of homeomorphisms whichfix x . By the theorem, Gx is generated by the Lorentz group anddilatations.

Therefore there are exactly four orbits under Gx :CL(x) \ x , CT (x) \ x , CS(x) \ x , the point x . 2



Second Result

Corollary


Proof:For x ∈ M, let Gx be the group of homeomorphisms whichfix x . By the theorem, Gx is generated by the Lorentz group anddilatations. Therefore there are exactly four orbits under Gx :CL(x) \ x , CT (x) \ x , CS(x) \ x , the point x . 2



Assumed Einstein’s Two Postulates.

Bondi’s k-factor.

Gamma factor γ(v) =1√

1− (v/c)2.

Lorentz Boost LV and Lorentz Transformation L.

Invariance of the Interval.

Inner Product g .

Predictions (simultaneity, Lorentz Contraction, mass, etc.).

Cones.

Defined Lorentz group L and L+.

Definined fine topology T F on M.

All mappings considered are in the Lorentz group.




Bondi’s k-factor.


1− (v/c)2.



Inner Product g .


Cones.







Bondi’s k-factor.


1− (v/c)2.



Inner Product g .


Cones.







Bondi’s k-factor.


1− (v/c)2.



Inner Product g .


Cones.







Bondi’s k-factor.


1− (v/c)2.



Inner Product g .


Cones.







Bondi’s k-factor.


1− (v/c)2.



Inner Product g .


Cones.







Bondi’s k-factor.


1− (v/c)2.



Inner Product g .


Cones.







Bondi’s k-factor.


1− (v/c)2.



Inner Product g .


Cones.







Bondi’s k-factor.


1− (v/c)2.



Inner Product g .


Cones.







Bondi’s k-factor.


1− (v/c)2.



Inner Product g .


Cones.







Bondi’s k-factor.


1− (v/c)2.



Inner Product g .


Cones.







Bondi’s k-factor.


1− (v/c)2.



Inner Product g .


Cones.






References

1. Bell, E. T. Men of Mathematics. Simon and Schuster: 1937.

2. Callahan, James J.The Geometry of Spacetime: An Introduction to Specialand General Relativity. Springer-Verlag: 2000.

3. Jackson, John David. Classical Electrodynamics. John Wiley& Sons Inc.: 1966.

4. Shadowitz, A. Special Relativity. W. B. Saunders Co.: 1986.

5. Taylor, Edwin F. and John Archibald Wheeler.Spacetime Physics. W. H. Freeman Co.: 1963.

6. Woodhouse, N. M. J. Special Relativity. SpringerUndergraduate Mathematics Series: 2003.

5. Zeeman, E. C. The topology of Minkowski space. TopologyVol. 6 pp. 161-170: 1967.


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