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Essentials of Relativity The Space of Relativity

The Mathematics of Special Relativity

Jared Ruiz Advised by Dr. Steven Kent May 6, 2009

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

Essentials of Relativity The Space of Relativity

Table of Contents

Essentials of Relativity

History Important Definitions Time Dilation Lorentz Transformation

Space of Relativity

Interval Predictions Topology

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

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History

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

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History

1632: Galileo Galilei publishes Dialogue concerning the two chief world systems

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History

1632: Galileo Galilei publishes Dialogue concerning the two chief world systems 1687: Isaac Newton releases Philosophiae Naturalis Principia Mathematica.

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History

1632: Galileo Galilei publishes Dialogue concerning the two chief world systems 1687: Isaac Newton releases Philosophiae Naturalis Principia Mathematica. 1865: James Clerk Maxwell unifies the theories of electricity and magnetism.

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History

1632: Galileo Galilei publishes Dialogue concerning the two chief world systems 1687: Isaac Newton releases Philosophiae Naturalis Principia Mathematica. 1865: James Clerk Maxwell unifies the theories of electricity and magnetism. ∇ · D = 4πρ ∇ · B = 0 ∇ × H = 4π c J + 1 c ∂D ∂t ∇ × E + 1 c ∂B ∂t = 0

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1887: Null result of the Michelson-Morley Experiment.

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1887: Null result of the Michelson-Morley Experiment. 1905: Albert Einstein reveals the Theory of Special Relativity.

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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 is

• bservable.

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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 is

• bservable.

Postulate 2 The velocity of light c is independent of the motion of the source.

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Definitions

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

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Definitions

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

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Definitions

Definition The four-dimensional set of axes is known as spacetime. Definition In classical mechanics, motion is described in a frame of reference.

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Definition A reference frame is inertial if every test particle initially at rest remains at rest and every particle in motion remains in motion without changing speed or direction.

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Definition A reference frame is inertial if every test particle initially at rest remains at rest and every particle in motion remains in motion without changing speed or direction. Definition An event is a point (t, x, y, z) in spacetime.

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Definition A reference frame is inertial if every test particle initially at rest remains at rest and every particle in motion remains in motion without 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 object will be called a world-line.

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Definition A reference frame is inertial if every test particle initially at rest remains at rest and every particle in motion remains in motion without 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 object will be called a world-line. Definition An observer is a series of clocks in an inertial reference frame which measures the time.

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Theorem Given two observers, O and O′, events in the inertial frames of reference 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 to O.

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Theorem Given two observers, O and O′, events in the inertial frames of reference 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 to O. This theorem is incorrect (physically). We have to reexamine our ideas of time.

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Definition The proper time τ along the worldline of a particle in constant motion is the time measured in an inertial coordinate system in which the particle is at rest.

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Definition The proper time τ along the worldline of a particle in constant motion is the time measured in an inertial coordinate system in which the particle is at rest. If we assume Postulate 2 by Einstein (that c is a constant), then t = kt′.

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Definition The proper time τ along the worldline of a particle in constant motion is the time measured in an inertial coordinate system in which the particle is at rest. If we assume Postulate 2 by Einstein (that c is a constant), then t = kt′. But if O′ is at rest relative to O, t′ = kt.

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Definition The proper time τ along the worldline of a particle in constant motion is the time measured in an inertial coordinate system in which the particle is at rest. If we assume Postulate 2 by Einstein (that c is a constant), then t = kt′. But if O′ is at rest relative to O, t′ = kt. This k is known as Bondi’s k-factor.

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Bondi’s k-factor

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Bondi’s k-factor

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Bondi’s k-factor

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

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Bondi’s k-factor

According to O, t = kt′, where k is a constant. According to O′, t′ = kt. tB and dB can be expressed as: dB = 1 2c(k2 − 1)t

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Bondi’s k-factor

According to O, t = kt′, where k is a constant. According to O′, t′ = kt. tB and dB can be expressed as: dB = 1 2c(k2 − 1)t and tB = 1 2(k2 + 1)t.

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The Gamma Factor

v = dB tB =

1 2c(k2 − 1)t 1 2(k2 + 1)t = c(k2 − 1)

(k2 + 1) .

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The Gamma Factor

v = dB tB =

1 2c(k2 − 1)t 1 2(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.

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The Gamma Factor

v = dB tB =

1 2c(k2 − 1)t 1 2(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′ = tB kt = (k2 + 1)t 2kt = γ(v)

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The Gamma Factor

v = dB tB =

1 2c(k2 − 1)t 1 2(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′ = tB kt = (k2 + 1)t 2kt = γ(v) γ(v) = (k2 + 1)t 2kt = 1

• 1 − v2/c2.

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The Lorentz Transformation

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The Lorentz Transformation

Theorem The inertial coordinate systems set up by two observers are related by:        t = t′ + (vx′/c2)

• 1 − (v/c)2

x = x′ + vt′

• 1 − (v/c)2

We can write this more concisely as ct x

• = γ(v)
• 1

v/c v/c 1 ct′ x′

• (1)

where v is the relative velocity.

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Proof

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Proof

First consider two observers moving at constant speeds.

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Proof

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

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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 ct x

• = 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 ✷.

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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 ct x

• = 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 ✷.

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Velocity Boost

In four dimensions,     ct x y z     =     γ γv/c γv/c γ 1 1         ct′ x′ y′ z′     (2) where v is the relative velocity and γ = γ(v).

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Velocity Boost

In four dimensions,     ct x y z     =     γ γv/c γv/c γ 1 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 observer would judge a moving particle’s velocity.

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

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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

• .

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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 .

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w = v + u 1 + uv/c2 . Letting |u| < c and |v| < c, we see that |w| < c since

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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.

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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. ✷

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In four-dimensions, we can say that

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In four-dimensions, we can say that     ct x y z     = L     ct′ x′ y′ z′     (3) where L = 1 H

• Lv

1 K T

• and H, K are 3 × 3
• rthogonal matrices.

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In four-dimensions, we can say that     ct x y z     = L     ct′ x′ y′ z′     (3) where L = 1 H

• Lv

1 K T

• and H, K are 3 × 3
• rthogonal matrices.

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

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In four-dimensions, we can say that     ct x y z     = L     ct′ x′ y′ z′     (3) where L = 1 H

• Lv

1 K T

• and H, K are 3 × 3
• rthogonal matrices.

Definition The matrix L in (3) is a Lorentz transformation if L−1 = gLTg, where g =     1 −1 −1 −1    .

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In four-dimensions, we can say that     ct x y z     = L     ct′ x′ y′ z′     (3) where L = 1 H

• Lv

1 K T

• and H, K are 3 × 3
• rthogonal matrices.

Definition The matrix L in (3) is a Lorentz transformation if L−1 = gLTg, where g =     1 −1 −1 −1    . L is orthochronous if l1,1 > 0, where l1,1 is the first entry of the first row of L

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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, and R3 = 0 × R3 the space axes.

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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, and R3 = 0 × R3 the space axes. The distance D from a point to the origin is D =

• x2 + y2 + z2.

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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, and R3 = 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.

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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, and R3 = 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.

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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, and R3 = 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.

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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, and R3 = 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 events x = (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.

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Invariance of the Interval

If c2(t2 − t1)2 − (x2 − x1)2 − (y2 − y1)2 − (z2 − z1)2 = 0 for an observer O,

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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.

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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.

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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 if X = LX ′ where L is the general Lorentz transformation.

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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 if X = LX ′ where L is the general Lorentz transformation. For x = (ct1, x1, y1, z1), y = (ct2, x2, y2, z2) ∈ M, the displacement four-vector X = y − x = c(t2 − t1) + (x2 − x1) + (y2 − y1) + (z2 − z1).

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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 if X = LX ′ where L is the general Lorentz transformation. For x = (ct1, x1, y1, z1), y = (ct2, x2, y2, z2) ∈ M, the displacement four-vector X = 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.

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Predictions

Definition The four-velocity of a particle, (V0, V1, V2, V3), is given by: V0 = c dt dτ , V1 = dx dτ , V2 = dy dτ , V3 = dz dτ .

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Predictions

Definition The four-velocity of a particle, (V0, V1, V2, V3), is given by: V0 = c dt dτ , V1 = dx dτ , V2 = dy dτ , V3 = dz dτ . Theorem Let an observer O be moving with constant velocity V . Then O sees two events A and B as being simultaneous if and only if the displacement g(X, V ) = 0, where X = B − A.

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Predictions

Definition The four-velocity of a particle, (V0, V1, V2, V3), is given by: V0 = c dt dτ , V1 = dx dτ , V2 = dy dτ , V3 = dz dτ . Theorem Let an observer O be moving with constant velocity V . Then O sees two events A and B as being simultaneous if and only if the displacement g(X, V ) = 0, where X = B − A. Corollary Two events which are simultaneous for one observer may not be simultaneous for another observer moving at constant velocity with the respect to the first observer.

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The Lorentz Contraction

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

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The Lorentz Contraction

Theorem If a rod has length R0 in a rest frame, then in an inertial coordinate system oriented in the direction of the unit vector e and moving with 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.

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

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Definition The rest mass m0 = m(0) of a body is the mass of a body measured in an inertial coordinate system in which the body is at rest. Theorem A body’s inertial mass m is a function of v, and can be rewritten as m = m(v) = γ(v)m0.

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Definition The rest mass m0 = m(0) of a body is the mass of a body measured in an inertial coordinate system in which the body is at rest. Theorem A body’s inertial mass m is a function of v, and can be rewritten as m = m(v) = γ(v)m0. Theorem E = mc2.

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Back to the Inner Product

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Back to the Inner Product

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

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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:

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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).

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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).

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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.

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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.

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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.

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Definition For an event x ∈ M, we have:

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Definition For an event x ∈ M, we have: Light-Cone C L(x) = {y : g(X, X) = 0}

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Definition For an event x ∈ M, we have: Light-Cone C L(x) = {y : g(X, X) = 0} Time-Cone C T(x) = {y : g(X, X) > 0}

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Definition For an event x ∈ M, we have: Light-Cone C L(x) = {y : g(X, X) = 0} Time-Cone C T(x) = {y : g(X, X) > 0} Space-Cone C S(x) = {y : g(X, X) < 0} where y ∈ M, and X is the displacement vector from x to y

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

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Definition For events x, y ∈ M we define a partial ordering < on M by x < y if the displacement vector X from x to y lies in the future light-cone That is, x < y if ty > tx, and g(X, X) > 0.

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Definition The matrix L in (3) is a Lorentz transformation if L−1 = gLTg, where g =     1 −1 −1 −1    .

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Definition The matrix L in (3) is a Lorentz transformation if L−1 = gLTg, where g =     1 −1 −1 −1    . Definition Define the Lorentz group L = {λ : M → M : ∀X, Y ∈ M, g(λX, λY ) = g(X, Y )}

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Definition The matrix L in (3) is a Lorentz transformation if L−1 = gLTg, where g =     1 −1 −1 −1    . Definition Define the Lorentz group L = {λ : M → M : ∀X, Y ∈ M, g(λX, λY ) = g(X, Y )} g(X ′, Y ′)

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Definition The matrix L in (3) is a Lorentz transformation if L−1 = gLTg, where g =     1 −1 −1 −1    . Definition Define the Lorentz group L = {λ : M → M : ∀X, Y ∈ M, g(λX, λY ) = g(X, Y )} g(X ′, Y ′) = g(LX, LY )

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Definition The matrix L in (3) is a Lorentz transformation if L−1 = gLTg, where g =     1 −1 −1 −1    . Definition Define the Lorentz group L = {λ : M → M : ∀X, Y ∈ M, g(λX, λY ) = g(X, Y )} g(X ′, Y ′) = g(LX, LY ) = g(λX, λY )

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Definition The matrix L in (3) is a Lorentz transformation if L−1 = gLTg, where g =     1 −1 −1 −1    . Definition Define the Lorentz group L = {λ : M → M : ∀X, Y ∈ M, g(λX, λY ) = g(X, Y )} g(X ′, Y ′) = g(LX, LY ) = g(λX, λY ) = g(X, Y ).

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Definition The matrix L in (3) is a Lorentz transformation if L−1 = gLTg, where g =     1 −1 −1 −1    . L is orthochronous if l1,1 > 0, where l1,1 is the first entry of the first row of L. Definition Define the Lorentz group L = {λ : M → M : ∀X, Y ∈ M, g(λX, λY ) = g(X, Y )} g(X ′, Y ′) = g(LX, LY ) = g(λX, λY ) = g(X, Y ).

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Definition The matrix L in (3) is a Lorentz transformation if L−1 = gLTg, where g =     1 −1 −1 −1    . L is orthochronous if l1,1 > 0, where l1,1 is the first entry of the first row of L. Definition Define the Lorentz group L = {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 whose elements preserve the partial ordering < on M.

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

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Usually we think of the topology on R4 as the standard Euclidean topology 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 spacelike events).

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

Essentials of Relativity The Space of Relativity

Usually we think of the topology on R4 as the standard Euclidean topology 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 spacelike events).

• 2. The group of all homeomorphisms of R4 include mappings of

no physical significance.

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

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Usually we think of the topology on R4 as the standard Euclidean topology 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 spacelike events).

• 2. The group of all homeomorphisms of R4 include mappings of

no physical significance. Two properties which T F will satisfy are: 1*. T F is not locally homogenous, and the light cone through any point can be deduced from T F.

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

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Usually we think of the topology on R4 as the standard Euclidean topology 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 spacelike events).

• 2. The group of all homeomorphisms of R4 include mappings of

no physical significance. Two properties which T F will satisfy are: 1*. T F is not locally homogenous, and the light cone through any point can be deduced from T F. 2*. The group of all homeomorphisms of the fine topology is generated by the inhomogeneous Lorentz group and dilatations.

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

Essentials of Relativity The Space of Relativity

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

Essentials of Relativity The Space of Relativity

Definition The fine topology T F of M induces the 1-dimensional Euclidean topology on gR and the 3-dimensional topology on gR3, where g ∈ G.

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

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Definition The fine topology T F of M induces the 1-dimensional Euclidean topology on gR and the 3-dimensional topology on gR3, where g ∈ G. A set U ∈ M is open in T F ⇐ ⇒ U ∩ gR is open in gR, and U ∩ gR3 is open in gR3. The ǫ-neighborhoods in T are NE

ǫ (x) = {y : d(x, y) < ǫ}.

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Definition The fine topology T F of M induces the 1-dimensional Euclidean topology on gR and the 3-dimensional topology on gR3, where g ∈ G. A set U ∈ M is open in T F ⇐ ⇒ U ∩ gR is open in gR, and U ∩ gR3 is open in gR3. The ǫ-neighborhoods in T are NE

ǫ (x) = {y : d(x, y) < ǫ}.

Definition We define the ǫ-neighborhoods in T F as NF

ǫ (x) = NE ǫ (x) ∩

• C T(x) ∪ C S(x)
• where x ∈ M.

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Theorem NF

ǫ (x) is open in T F.

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Theorem NF

ǫ (x) is open in T F.

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

ǫ (x) ∩ A =

• Jared RuizAdvised by Dr. Steven Kent

The Mathematics of Special Relativity

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Theorem NF

ǫ (x) is open in T F.

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

ǫ (x) ∩ A =

• NE

ǫ (x) ∩ A

if x ∈ A

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Theorem NF

ǫ (x) is open in T F.

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

ǫ (x) ∩ A =

• NE

ǫ (x) ∩ A

if x ∈ A

• NE

ǫ (x) \ C L(x)

• ∩ A

if x / ∈ A

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Theorem NF

ǫ (x) is open in T F.

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

ǫ (x) ∩ A =

• NE

ǫ (x) ∩ A

if x ∈ A

• NE

ǫ (x) \ C L(x)

• ∩ A

if x / ∈ A

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Theorem NF

ǫ (x) is open in T F.

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

ǫ (x) ∩ A =

• NE

ǫ (x) ∩ A

if x ∈ A

• NE

ǫ (x) \ C L(x)

• ∩ A

if x / ∈ A

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First Result

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

Essentials of Relativity The Space of Relativity

First Result

Theorem The group of all homeomorphisms of T F is G.

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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 a homeomorphism.

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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 a

• homeomorphism. Now we have to show every homeomorphism

h :

• M, T F

→

• M, T F

is in G.

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

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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 a

• homeomorphism. Now we have to show every homeomorphism

h :

• 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.

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

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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 a

• homeomorphism. Now we have to show every homeomorphism

h :

• 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 element

• f G0.

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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 a

• homeomorphism. Now we have to show every homeomorphism

h :

• 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 element

• f G0.

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

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Second Result

Corollary The light, time, and space cones through a point can be deduced from the topology.

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Second Result

Corollary The light, time, and space cones through a point can be deduced from the topology. Proof:For x ∈ M, let Gx be the group of homeomorphisms which fix x.

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Second Result

Corollary The light, time, and space cones through a point can be deduced from the topology. Proof:For x ∈ M, let Gx be the group of homeomorphisms which fix x. By the theorem, Gx is generated by the Lorentz group and dilatations.

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Second Result

Corollary The light, time, and space cones through a point can be deduced from the topology. Proof:For x ∈ M, let Gx be the group of homeomorphisms which fix x. By the theorem, Gx is generated by the Lorentz group and

• dilatations. Therefore there are exactly four orbits under Gx:

C L(x) \ x, C T(x) \ x, C S(x) \ x, the point x. ✷

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Assumed Einstein’s Two Postulates.

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Assumed Einstein’s Two Postulates. Bondi’s k-factor.

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Assumed Einstein’s Two Postulates. Bondi’s k-factor. Gamma factor γ(v) = 1

• 1 − (v/c)2 .

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

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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.

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

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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.

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

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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.

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

Essentials of Relativity The Space of Relativity

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.).

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

Essentials of Relativity The Space of Relativity

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.

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

Essentials of Relativity The Space of Relativity

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+.

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

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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.

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

Essentials of Relativity The Space of Relativity

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.

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References

• 1. Bell, E. T. Men of Mathematics. Simon and Schuster: 1937.
• 2. Callahan, James J.

The Geometry of Spacetime: An Introduction to Special and 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. Springer

Undergraduate Mathematics Series: 2003.

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

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