Lorentz Transformation − Eφ+E Φ Cosh Φ := 2 Φ− −Φ Sinh Φ := E E Cosh Φ Sinh Φ 2 Λ(Φ) := − W:Hyperbolic Function Sinh Φ Cosh Φ −
Special relativity and steps towards general relativity: SR
(c) CC-BY-SA-3.0 SR+GR SR+GR: construct spacetime from set theory •
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.1 SR+GR SR+GR: construct spacetime from set theory • start with X = set of points, no distances between points,• angles, etc. defined
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.1 SR+GR SR+GR: construct spacetime from set theory • start with X = set of points, no distances between points,• angles, etc. defined no absolute simultaneity = “time is not absolute" •
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.1 SR+GR SR+GR: construct spacetime from set theory • start with X = set of points, no distances between points,• angles, etc. defined no absolute simultaneity = “time is not absolute" • add properties to X that satisfy theorems •
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.1 SR+GR SR+GR: construct spacetime from set theory • start with X = set of points, no distances between points,• angles, etc. defined no absolute simultaneity = “time is not absolute" • add properties to X that satisfy theorems • differentiable 4-(pseudo-)manifold •→
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.1 SR+GR SR+GR: construct spacetime from set theory • start with X = set of points, no distances between points,• angles, etc. defined no absolute simultaneity = “time is not absolute" • add properties to X that satisfy theorems • differentiable 4-(pseudo-)manifold •→ point particle in space w:World line in spacetime • →
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.1 SR+GR SR+GR: construct spacetime from set theory • start with X = set of points, no distances between points,• angles, etc. defined no absolute simultaneity = “time is not absolute" • add properties to X that satisfy theorems • differentiable 4-(pseudo-)manifold •→ point particle in space w:World line in spacetime • → point in spacetime spacetime “event" • →
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.1 SR+GR SR+GR: construct spacetime from set theory • start with X = set of points, no distances between points,• angles, etc. defined no absolute simultaneity = “time is not absolute" • add properties to X that satisfy theorems • differentiable 4-(pseudo-)manifold •→ point particle in space w:World line in spacetime • → point in spacetime spacetime “event" • → SR: spacetime = w:Minkowski space •
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.1 SR+GR SR+GR: construct spacetime from set theory • start with X = set of points, no distances between points,• angles, etc. defined no absolute simultaneity = “time is not absolute" • add properties to X that satisfy theorems • differentiable 4-(pseudo-)manifold •→ point particle in space w:World line in spacetime • → point in spacetime spacetime “event" • → SR: spacetime = w:Minkowski space • GR: spacetime = a solution of the w:Einstein• field equations
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.1 SR: Minkowski spacetime y p (x,y)
d
O x (0,0)
p at (x,y), distance from observer at O is d
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.2 SR: Minkowski spacetime y' p p ′ x = (x',y') py′
cos θ sin θ px d sin θ cos θ py − x'
O (0,0)
p at (x′,y′), distance from observer at O is d = unchanged
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.2 SR: Minkowski spacetime t p (x,t)
s ∆
O x (0,0)
p at (x,t), w:invariant interval from observer at O is ∆s where (∆s)2 = (∆t)2 + (∆x)2 −
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.2 SR: Minkowski spacetime
t' p p ′ x = (x',t') pt′ cosh φ sinh φ px s − ∆ sinh φ cosh φ p − t x' O (0,0)
p at (x′,t′), invariant interval from observer at O is ∆s = (∆s)2 = (∆t′)2 + (∆x′)2 = unchanged −
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.2 SR: Lorentz transformation − eφ+e φ cosh φ := 2 φ− −φ sinh φ := e e cosh φ sinh φ 2 Λ(φ) := − w:hyperbolic function sinh φ cosh φ −
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.3 SR: Lorentz transformation − eφ+e φ cosh φ := 2 φ− −φ sinh φ := e e cosh φ sinh φ 2 Λ(φ) := − w:hyperbolic function sinh φ cosh φ −
p ′ p x = Λ x pt′ pt
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.3 SR: Lorentz transformation − eφ+e φ cosh φ := 2 φ− −φ sinh φ := e e cosh φ sinh φ 2 Λ(φ) := − w:hyperbolic function sinh φ cosh φ −
p ′ p x = Λ x pt′ pt units: Λ only makes sense if same units for x, t, x′, t′
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.3 SR: Lorentz transformation − eφ+e φ cosh φ := 2 φ− −φ sinh φ := e e cosh φ sinh φ 2 Λ(φ) := − w:hyperbolic function sinh φ cosh φ −
p ′ p x = Λ x pt′ pt units: Λ only makes sense if same units for x, t, x′, t′ Definition: 1 m := (1/2.99792458 108)s 10−8.5 s 3 ns × ≈ ≈
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.3 SR: Lorentz transformation − eφ+e φ cosh φ := 2 φ− −φ sinh φ := e e cosh φ sinh φ 2 Λ(φ) := − w:hyperbolic function sinh φ cosh φ −
p ′ p x = Λ x pt′ pt units: Λ only makes sense if same units for x, t, x′, t′ Definition: 1 m := (1/2.99792458 108)s 10−8.5 s 3 ns × ≈ ≈ 2.99792458×108 m speed of light in a vacuum = c = 1 s
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.3 SR: Lorentz transformation − eφ+e φ cosh φ := 2 φ− −φ sinh φ := e e cosh φ sinh φ 2 Λ(φ) := − w:hyperbolic function sinh φ cosh φ −
p ′ p x = Λ x pt′ pt units: Λ only makes sense if same units for x, t, x′, t′ Definition: 1 m := (1/2.99792458 108)s 10−8.5 s 3 ns × ≈ ≈ 2.99792458×108 m speed of light in a vacuum = c = 1 s 2.99792458×108×(1/2.99792458×108) s = 1 s
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.3 SR: Lorentz transformation − eφ+e φ cosh φ := 2 φ− −φ sinh φ := e e cosh φ sinh φ 2 Λ(φ) := − w:hyperbolic function sinh φ cosh φ −
p ′ p x = Λ x pt′ pt units: Λ only makes sense if same units for x, t, x′, t′ Definition: 1 m := (1/2.99792458 108)s 10−8.5 s 3 ns × ≈ ≈ 2.99792458×108 m speed of light in a vacuum = c = 1 s 1 s = 1 s
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.3 SR: Lorentz transformation − eφ+e φ cosh φ := 2 φ− −φ sinh φ := e e cosh φ sinh φ 2 Λ(φ) := − w:hyperbolic function sinh φ cosh φ −
p ′ p x = Λ x pt′ pt units: Λ only makes sense if same units for x, t, x′, t′ Definition: 1 m := (1/2.99792458 108)s 10−8.5 s 3 ns × ≈ ≈ 2.99792458×108 m speed of light in a vacuum = c = 1 s 1 s = 1 s = 1 (dimensionless)
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.3 SR: rapidity φ vs velocity β What is φ ?
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.4 SR: rapidity φ vs velocity β What is φ ?
observer A has worldline (x,t)=(0,t)
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.4 SR: rapidity φ vs velocity β What is φ ?
observer B has worldline (x′,t′)=(0,t′)
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.4 SR: rapidity φ vs velocity β What is φ ?
observer B has worldline (x′,t′)=(0,t′) in A’s coordinate system, B’s worldline is:
0 x ′ = Λ t t
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.4 SR: rapidity φ vs velocity β What is φ ?
observer B has worldline (x′,t′)=(0,t′) in A’s coordinate system, B’s worldline is:
′ x −1 0 t sinh φ = Λ ′ = ′ ⇒ t t t cosh φ
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.4 SR: rapidity φ vs velocity β What is φ ?
observer B has worldline (x′,t′)=(0,t′) in A’s coordinate system, B’s worldline is:
′ x −1 0 t sinh φ = Λ ′ = ′ ⇒ t t t cosh φ
x = t′ sinh φ ⇒
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.4 SR: rapidity φ vs velocity β What is φ ?
observer B has worldline (x′,t′)=(0,t′) in A’s coordinate system, B’s worldline is:
′ x −1 0 t sinh φ = Λ ′ = ′ ⇒ t t t cosh φ
x = t′ sinh φ = t′ cosh φ sinh φ ⇒ cosh φ
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.4 SR: rapidity φ vs velocity β What is φ ?
observer B has worldline (x′,t′)=(0,t′) in A’s coordinate system, B’s worldline is:
′ x −1 0 t sinh φ = Λ ′ = ′ ⇒ t t t cosh φ
x = t′ sinh φ = t′ cosh φ tanh φ ⇒
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.4 SR: rapidity φ vs velocity β What is φ ?
observer B has worldline (x′,t′)=(0,t′) in A’s coordinate system, B’s worldline is:
′ x −1 0 t sinh φ = Λ ′ = ′ ⇒ t t t cosh φ
x = t′ sinh φ = t tanh φ ⇒
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.4 SR: rapidity φ vs velocity β What is φ ?
observer B has worldline (x′,t′)=(0,t′) in A’s coordinate system, B’s worldline is:
′ x −1 0 t sinh φ = Λ ′ = ′ ⇒ t t t cosh φ
x = t′ sinh φ = t tanh φ = βt ⇒ where velocity β := v/c v = tanh φ ≡
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.4 SR: calibration Where does (x′,t′)=(0, 1) lie for observer A?
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.5 SR: calibration Where does (x′,t′)=(0, 1) lie for observer A? Where does (x′,t′)=(1, 0) lie for observer A?
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.5 SR: calibration Where does (x′,t′)=(0, 1) lie for observer A? Where does (x′,t′)=(1, 0) lie for observer A?
y y' 1 1
d = 1 constant distance x' d = 1 from (0, 0) 1 O x (0,0) 1
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.5 SR: calibration Where does (x′,t′)=(0, 1) lie for observer A? Where does (x′,t′)=(1, 0) lie for observer A?
t t' constant interval φ (∆s)2 = 1 from (0, 0) cosh 1 − 1 1 − 0 − 1 = Λ = 2 ) 1 s timelike interval (∆ sinh φ x' cosh φ O x (0,0) sinh φ
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.5 SR: calibration Where does (x′,t′)=(0, 1) lie for observer A? Where does (x′,t′)=(1, 0) lie for observer A?
t t'
1 1 − = 1 2 ) constant interval s 2 (∆ timelike interval from 2 = +1 (∆s) = +1 (0, 0) s) (∆ −1 1 cosh φ spacelike interval Λ = 1 x' 0 sinh φ O x (0,0) 1
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.5 SR: calibration Where does (x′,t′)=(0, 1) lie for observer A? Where does (x′,t′)=(1, 0) lie for observer A? Can high φ push the t′ axis close to the x axis?
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.5 SR: calibration Where does (x′,t′)=(0, 1) lie for observer A? Where does (x′,t′)=(1, 0) lie for observer A? Can high φ push the t′ axis close to the x axis?
t t′′ 1
1 x′′ 1 − = 1 2 ) s timelike interval (∆
2 = +1spacelike interval s) O (∆ x (0,0) 1
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.5 SR: calibration Where does (x′,t′)=(0, 1) lie for observer A? Where does (x′,t′)=(1, 0) lie for observer A? Can high φ push the t′ axis close to the x axis?
1.5
1
0.5
φ
0 = tanh β
-0.5
-1
-1.5
-4 -2 0 2 4 φ
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.5 SR: effect of Λ on x = t (photons) What happens to a photon under Lorentz transformation?
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.6 SR: effect of Λ on x = t (photons) What happens to a photon under Lorentz transformation?
0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.6 SR: effect of Λ on x = t (photons) What happens to a photon under Lorentz transformation? Observer A: photon worldline is the set of spacetime events (t,t) t 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.6 SR: effect of Λ on x = t (photons) What happens to a photon under Lorentz transformation? Observer A: photon worldline is the set of spacetime events (t,t) t 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.6 SR: effect of Λ on x = t (photons) What happens to a photon under Lorentz transformation? Observer A: photon worldline is the set of spacetime events (t,t) t 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.6 SR: effect of Λ on x = t (photons) What happens to a photon under Lorentz transformation? Observer A: photon worldline is the set of spacetime events (t,t) t 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.6 SR: effect of Λ on x = t (photons) What happens to a photon under Lorentz transformation? Observer A: photon worldline is the set of spacetime events (t,t) t 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.6 SR: effect of Λ on x = t (photons) What happens to a photon under Lorentz transformation? Observer A: photon worldline is the set of spacetime events (t,t) t 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.6 SR: adding velocities interstellar ark travels at β1 = 0.1 from Sun, sends out rocket at β2 = 0.5; rocket’s speed β3 in Sun frame =? 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.7 SR: adding velocities interstellar ark travels at β1 = 0.1 from Sun, sends out rocket at β2 = 0.5; rocket’s speed β3 in Sun frame =? t t′ t′′ 1 1 1 x′′ x′ = ′ t cosh − sinh φ1 φ1 x O − sinh cosh φ1 φ1 t (0,0) x′ where tanh φ1 = β1 = 0.1 x 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.7 SR: adding velocities interstellar ark travels at β1 = 0.1 from Sun, sends out rocket at β2 = 0.5; rocket’s speed β3 in Sun frame =? t t′ t′′ 1 1 1 x′′ x′′ = ′′ t cosh − sinh ′ φ2 φ2 x O − sinh cosh ′ φ2 φ2 t (0,0) x′ where tanh φ2 = β2 = 0.5 x 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.7 SR: adding velocities interstellar ark travels at β1 = 0.1 from Sun, sends out rocket at β2 = 0.5; rocket’s speed β3 in Sun frame =? t t′ t′′ 1 1 1 x ′′ x′′ = ′′ t cosh − sinh ′ x′ φ2 φ2 x ′ O − sinh φ2 cosh φ2 t x (0,0) where tanh φ2 = β2 = 0.5 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.7 SR: adding velocities interstellar ark travels at β1 = 0.1 from Sun, sends out rocket at β2 = 0.5; rocket’s speed β3 in Sun frame =? t t′ t′′ 1 1 1 x′′ ′′ x′ x x O = Λ(φ2)Λ(φ1) x ′′ (0,0) t t 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.7 SR: adding velocities interstellar ark travels at β1 = 0.1 from Sun, sends out rocket at β2 = 0.5; rocket’s speed β3 in Sun frame =? t t′ t′′ 1 1 1 x′′ ′′ x′ x x O = Λ(φ2)Λ(φ1) x ′′ (0,0) t t but Λ(φ2)Λ(φ1)=Λ(φ1 + φ2) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.7 SR: adding velocities interstellar ark travels at β1 = 0.1 from Sun, sends out rocket at β2 = 0.5; rocket’s speed β3 in Sun frame =? t t′ t′′ 1 1 1 x′′ ′′ x′ x x O = Λ(φ2)Λ(φ1) x ′′ (0,0) t t but Λ(φ2)Λ(φ1)=Λ(φ1 + φ2) cf. rotation θ1 “plus" rotation θ2 = rotation θ1 + θ2 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.7 SR: adding velocities interstellar ark travels at β1 = 0.1 from Sun, sends out rocket at β2 = 0.5; rocket’s speed β3 in Sun frame =? t t′ t′′ 1 1 1 x′′ ′′ x′ x x O = Λ(φ2)Λ(φ1) x ′′ (0,0) t t but Λ(φ2)Λ(φ1)=Λ(φ1 + φ2) so β3 = tanh (φ1 + φ2) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.7 SR: adding velocities interstellar ark travels at β1 = 0.1 from Sun, sends out rocket at β2 = 0.5; rocket’s speed β3 in Sun frame =? t t′ t′′ 1 1 1 x′′ ′′ x′ x x O = Λ(φ2)Λ(φ1) x ′′ (0,0) t t but Λ(φ2)Λ(φ1)=Λ(φ1 + φ2) so β = tanh (φ + φ ) = tanh φ1+ tanh φ2 3 1 2 1+ tanh φ1 tanh φ2 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.7 SR: adding velocities interstellar ark travels at β1 = 0.1 from Sun, sends out rocket at β2 = 0.5; rocket’s speed β3 in Sun frame =? t t′ t′′ 1 1 1 x′′ ′′ x′ x x O = Λ(φ2)Λ(φ1) x ′′ (0,0) t t but Λ(φ2)Λ(φ1)=Λ(φ1 + φ2) so β = tanh (φ + φ ) = β1+β2 3 1 2 1+β1β2 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.7 SR: adding velocities interstellar ark travels at β1 = 0.1 from Sun, sends out rocket at β2 = 0.5; rocket’s speed β3 in Sun frame =? t t′ t′′ 1 1 1 x′′ ′′ x′ x x O = Λ(φ2)Λ(φ1) x ′′ (0,0) t t but Λ(φ2)Λ(φ1)=Λ(φ1 + φ2) β1+β2 0.1+0.5 so β3 = tanh (φ1 + φ2) = = × 0.57 1+β1β2 1+0.1 0.5 ≈ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.7 SR: Lorentz factor Λ: alternative to hyperbolic trig functions 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.8 SR: Lorentz factor Λ: alternative to hyperbolic trig functions − eφ+e φ cosh φ := 2 φ− −φ sinh φ := e e cosh φ sinh φ 2 Λ(φ) := − w:hyperbolic function sinh φ cosh φ − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.8 SR: Lorentz factor Λ: alternative to hyperbolic trig functions β = tanh φ γ := (1 β2)−1/2 = Lorentz factor− γ = cosh φ γ βγ Λ(β) := − βγ = sinh φ βγ γ − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.8 SR: worldline time dilation t t′ φ 1 cosh 1 x′ O x (0,0) sinh φ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.9 SR: worldline time dilation t t′ φ 1 cosh 1 x′ O x (0,0) sinh φ cosh φ γ ≡ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.9 SR: worldline time dilation t t′ φ 1 cosh 1 x′ O x (0,0) sinh φ cosh φ γ 1 ≡ ≡ √1−β2 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.9 SR: worldline time dilation t t′ φ 1 cosh 1 x′ O x (0,0) sinh φ cosh φ γ 1 > 1 ≡ ≡ √1−β2 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.9 SR: worldline time dilation t t′ φ 1 cosh 1 worldline “time dilation" x′ O x (0,0) sinh φ cosh φ γ 1 > 1 ≡ ≡ √1−β2 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.9 SR: worldline time dilation t t′ φ 1 cosh 1 worldline “time dilation" x′ muons: mean lifetime 2197 ns 15 km O ≪ x (0,0) sinh φ cosh φ γ 1 > 1 ≡ ≡ √1−β2 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.9 SR: worldline time dilation t t′ φ 1 cosh 1 worldline “time dilation" x′ muons: mean lifetime 2197 ns 15 km O ≪ x (0,0) sinh φ time dilation muons cosh φ γ 1 > 1 can hit the ground⇒ ≡ ≡ √1−β2 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.9 SR: worldsheet space contraction t ′ t β sinh φ = β2γ r q x′ φ p 1 sinh O 1 x (0,0) cosh φ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.10 SR: worldsheet space contraction t ′ t β sinh φ = β2γ r q x′ φ p 1 sinh O 1 x (0,0) cosh φ ∆s2(q, p) = 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.10 SR: worldsheet space contraction t ′ t β sinh φ = β2γ r q x′ φ p 1 sinh O 1 x (0,0) cosh φ ∆s2(q, p) = cosh φ β sinh φ − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.10 SR: worldsheet space contraction t ′ t β sinh φ = β2γ r q x′ φ p 1 sinh O 1 x (0,0) cosh φ ∆s2(q, p) = γ ββγ − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.10 SR: worldsheet space contraction t ′ t β sinh φ = β2γ r q x′ φ p 1 sinh O 1 x (0,0) cosh φ ∆s2(q, p) = (1 β2)γ − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.10 SR: worldsheet space contraction t ′ t β sinh φ = β2γ r q x′ φ p 1 sinh O 1 x (0,0) cosh φ ∆s2(q, p) = (1 β2)1+(−1/2) − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.10 SR: worldsheet space contraction t ′ t β sinh φ = β2γ r q x′ φ p 1 sinh O 1 x (0,0) cosh φ ∆s2(q, p) = (1 β2)1/2 − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.10 SR: worldsheet space contraction t ′ t β sinh φ = β2γ r q x′ φ p 1 sinh O 1 x (0,0) cosh φ ∆s2(q, p) = γ−1 < 1 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.10 SR: worldsheet space contraction t ′ t β sinh φ = β2γ r q x′ φ p 1 sinh O 1 x (0,0) cosh φ ∆s2(q, p) = γ−1 < 1 worldsheet “space contraction" 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.10 SR: worldsheet space contraction t t′ r q (0,0) 1 p O x′ 1 β − Λ( φ) (1 − , x −β) T 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.11 SR: worldsheet space contraction t t′ r q (0,0) 1 p O x′ 1 β − Λ( φ) (1 − , x −β) T − 1 cosh φ β sinh φ Λ 1 = − β sinh φ β cosh φ − − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.11 SR: worldsheet space contraction t t′ r q (0,0) 1 p O x′ 1 β − Λ( φ) (1 − , x −β) T 2 − 1 cosh φ(1 β ) Λ 1 = − β cosh φ( tanh φ tanh φ) − − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.11 SR: worldsheet space contraction t t′ r q (0,0) 1 p O x′ 1 β − Λ( φ) (1 − , x −β) T 2 − 1 γ(1 β ) Λ 1 = − β 0 − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.11 SR: worldsheet space contraction t t′ r q (0,0) 1 p O x′ 1 β − Λ( φ) (1 − , x −β) T 1 γ−1 Λ−1 = β 0 − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.11 SR: worldsheet space contraction t t′ r q (0,0) 1 p O x′ 1 β − Λ( φ) (1 − , x −β) T ∆s2(q, p) = γ−1 < 1 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.11 SR: worldsheet space contraction t t′ r q (0,0) 1 p O x′ 1 β − Λ( φ) (1 − , x −β) T ∆s2(q, p) = γ−1 < 1 worldsheet “space contraction" 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.11 SR: Doppler shift t t′ photon 1 30 ps O 30 ps x (0,0) x′ (cosh φ + sinh φ)30 ps 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.12 SR: Doppler shift t t′ photon 1 30 ps O 30 ps x (0,0) x′ (cosh φ + sinh φ)30 ps see photon worldline calculation 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.12 SR: Doppler shift t t′ photon 1 30 ps O 30 ps x (0,0) x′ (cosh φ + sinh φ)30 ps see photon worldline calculation x′ = (cosh φ + sinh φ)t 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.12 SR: Doppler shift t t′ photon 1 30 ps O 30 ps x (0,0) x′ (cosh φ + sinh φ)30 ps see photon worldline calculation x′ = (cosh φ + sinh φ)x 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.12 SR: Doppler shift t t′ photon 1 30 ps O 30 ps x (0,0) x′ (cosh φ + sinh φ)30 ps see photon worldline calculation x′/x = cosh φ + sinh φ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.12 SR: Doppler shift t t′ photon 1 30 ps O 30 ps x (0,0) x′ (cosh φ + sinh φ)30 ps see photon worldline calculation x′/x = γ + βγ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.12 SR: Doppler shift t t′ photon 1 30 ps O 30 ps x (0,0) x′ (cosh φ + sinh φ)30 ps see photon worldline calculation x′/x = γ(1 + β) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.12 SR: Doppler shift t t′ photon 1 30 ps O 30 ps x (0,0) x′ (cosh φ + sinh φ)30 ps see photon worldline calculation x′/x = γ(1 + β) = 1+β √1−β2 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.12 SR: Doppler shift t t′ photon 1 30 ps O 30 ps x (0,0) x′ (cosh φ + sinh φ)30 ps see photon worldline calculation ′ √(1+β)2 x /x = γ(1 + β) = √(1−β)(1+β) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.12 SR: Doppler shift t t′ photon 1 30 ps O 30 ps x (0,0) x′ (cosh φ + sinh φ)30 ps see photon worldline calculation ′ 1+β x /x = γ(1 + β) = 1−β 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.12 SR: Doppler shift t t′ photon 1 30 ps O 30 ps x (0,0) x′ (cosh φ + sinh φ)30 ps see photon worldline calculation 1 + z := λ′/λ = γ(1 + β) 1+β redshift ≡ 1−β 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.12 SR: Doppler shift t t′ photon 1 30 ps O 30 ps x (0,0) x′ (cosh φ + sinh φ)30 ps see photon worldline calculation 1 + z := λ′/λ = γ(1 + β) 1+β redshift ≡ 1−β when β 1, z β ⇒ ≪ ≈ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.12 SR: relativistic aberration y (x,y) plane photonB θ sin θ x (0,0,0) cos θ (x′,y′) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.13 SR: relativistic aberration y (x,y) plane photonB θ sin θ x (0,0,0) cos θ (x′,y′) frame event B: (x,y,t) = (cos θ, sin θ, 1) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.13 SR: relativistic aberration event B: (x,y,t) = (cos θ, sin θ, 1) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.13 SR: relativistic aberration event B: (x,y,t) = (cos θ, sin θ, 1) cos θ −1 Λ sin θ 1 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.13 SR: relativistic aberration event B: (x,y,t) = (cos θ, sin θ, 1) cos θ γ 0 βγ cos θ −1 Λ sin θ = 0 1 0 sin θ 1 βγ 0 γ 1 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.13 SR: relativistic aberration event B: (x,y,t) = (cos θ, sin θ, 1) cos θ γ cos θ + βγ −1 Λ sin θ = sin θ 1 βγ cos θ + γ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.13 SR: relativistic aberration event B: (x,y,t) = (cos θ, sin θ, 1) cos θ γ(cos θ + β) −1 Λ sin θ = sin θ 1 γ(1 + β cos θ) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.13 SR: relativistic aberration event B: (x,y,t) = (cos θ, sin θ, 1) (x ,y ) plane y′ ′ ′ θ photonB sin θ′ =? x′ (0,0,0) γ(β + cos θ) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.13 SR: relativistic aberration event B: (x,y,t) = (cos θ, sin θ, 1) y (x,y) plane photonB θ sin θ x (0,0,0) cos θ (x′,y′) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.13 SR: relativistic aberration event B: (x,y,t) = (cos θ, sin θ, 1) t t′ photon γ 1+(1 = 0) β 1 θ cos θ ) (photon with O cos θ x (0,0) x′ γ(β + cos θ) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.13 SR: relativistic aberration event B: (x,y,t) = (cos θ, sin θ, 1) y (x,y) plane photonB θ sin θ x (0,0,0) cos θ (x′,y′) frame ′ sin θ tan θ = γ(β+cos θ) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.13 SR: relativistic aberration event B: (x,y,t) = (cos θ, sin θ, 1) y (x,y) plane photonB θ sin θ x (0,0,0) cos θ (x′,y′) frame ′ sin θ tan θ = γ(β+cos θ) < tan θ if 0 < β < 1 w:Relativistic aberration 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.13 SR: relativistic aberration event B: (x,y,t) = (cos θ, sin θ, 1) y (x,y) plane photonB θ sin θ x (0,0,0) cos θ (x′,y′) frame ′ sin θ tan θ = γ(β+cos θ) < tan θ if 0 < β < 1 w:Relativistic aberration relativistic beaming, e.g. AGN jets ⇒ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.13 SR: world line 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.14 SR: world line lightlike interval = null interval: (∆s)2 = 0 spacetime = 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.14 SR: world line lightlike interval = null interval: (∆s)2 = 0 spacetime = on past w:light cone + inside past light cone 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.14 SR: world line lightlike interval = null interval: (∆s)2 = 0 spacetime = on past w:light cone + inside past light cone + on future light cone + inside future light cone 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.14 SR: world line lightlike interval = null interval: (∆s)2 = 0 spacetime = on past w:light cone + inside past light cone + on future light cone + inside future light cone + elsewhere 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.14 SR: world line Lorentz transform of world line 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.15 SR: world line Lorentz transform of world line 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.15 SR: world line Lorentz transform of world line coordinate time in spacetime model = time in your brain• (thinking) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.15 SR: world line Lorentz transform of world line coordinate time in spacetime model = time in your brain• (thinking) dt can be positive or negative • dtthinking 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.15 SR: world line Lorentz transform of world line coordinate time in spacetime model = time in your brain• (thinking) dt dλ can be positive or negative, λ arbitrary real •parameter 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.15 SR: world line Lorentz transform of world line coordinate time in spacetime model = time in your brain• (thinking) dt dλ can be positive or negative, λ arbitrary real •parameter “elsewhere" spacetime events can change from past to • dt future even though dλ > 0 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.15 SR: world line Lorentz transform of world line coordinate time in spacetime model = time in your brain• (thinking) dt dλ can be positive or negative, λ arbitrary real •parameter “elsewhere" spacetime events can change from past to • dt future even though dλ > 0 w:proper time τ := time along a worldline measured by • clock following that worldline 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.15 SR: world line Lorentz transform of world line coordinate time in spacetime model = time in your brain• (thinking) dt dλ can be positive or negative, λ arbitrary real •parameter “elsewhere" spacetime events can change from past to • dt future even though dλ > 0 w:proper time τ := time along a worldline measured by • clock following that worldline often dτ is useful for integrating • 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.15 SR: Rietdijk–Putnam–Penrose p. 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.16 SR: Rietdijk–Putnam–Penrose p. b:Inertialoverlay.GIF each observer can synchronise clocks + rods • 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.16 SR: Rietdijk–Putnam–Penrose p. w:Rietdijk-Putnam argument b:Rel2.gif 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.16 SR: Rietdijk–Putnam–Penrose p. w:Rietdijk-Putnam argument b:Rel3.gif 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.16 SR: tachyons and causality t O x (0,0) observer “at rest" 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.17 SR: tachyons and causality t ) αt,t ) = ( x,t O tachyon ( x (0,0) add a tachyon with speed α > 1 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.17 SR: tachyons and causality t ) αt,t ) = ( x,t O tachyon ( x (0,0) add a tachyon with speed α > 1 choose rocket at speed β with 1/α < β < 1 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.17 SR: tachyons and causality t t ′ x′ ) αt,t ) = ( x,t O tachyon ( x (0,0) add a tachyon with speed α > 1 choose rocket at speed β with 1/α < β < 1 add axes x′,t′ for the rocket 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.17 SR: tachyons and causality t t ′ x′ ) αt,t ) = ( x,t O tachyon ( x (0,0) add a tachyon with speed α > 1 choose rocket at speed β with 1/α < β < 1 add axes x′,t′ for the rocket rocket frame: (αt,t) becomes Λ(αt,t)T 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.17 SR: tachyons and causality x′ αt ′ = Λ = t t 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.17 SR: tachyons and causality x′ αt γαt βγt = Λ = − t′ t αβγt + γt − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.17 SR: tachyons and causality x′ αt α β = Λ = γt − t′ t αβ + 1 − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.17 SR: tachyons and causality x′ αt α β = Λ = γt − t′ t αβ + 1 − x′ = γt(α β) > 0 since α > 1 > β − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.17 SR: tachyons and causality x′ αt α β = Λ = γt − t′ t αβ + 1 − x′ = γt(α β) > 0 since α > 1 > β − t′ = γt( αβ + 1) < 0 since we chose β > 1/α − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.17 SR: tachyons and causality x′ αt α β = Λ = γt − t′ t αβ + 1 − x′ = γt(α β) > 0 since α > 1 > β − t′ = γt( αβ + 1) < 0 since we chose β > 1/α − dt′/dt < 0 same sequence of spacetime events = tachyon worldline: t increases for observer “at rest", t′ decreases for rocket observer (with β > 1/α) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.17 SR: tachyons and causality x′ αt α β = Λ = γt − t′ t αβ + 1 − x′ = γt(α β) > 0 since α > 1 > β − t′ = γt( αβ + 1) < 0 since we chose β > 1/α − dt′/dt < 0 same sequence of spacetime events = tachyon worldline: t increases for observer “at rest", t′ decreases for rocket observer (with β > 1/α) observer at rest: tachyon emitted at origin • rocket: tachyon absorbed at origin • 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.17 SR: tachyonic antitelephone t ′ t ) ′C = ( L,t tC C = (αtC,tC) tachyon 0) x′ (L, ) B x L/α L, − (0, 0) A = ( 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.18 SR: tachyonic antitelephone t ′ t ) ′C = ( L,t tC C = (αtC,tC) tachyon 0) x′ (L, ) B x L/α L, − (0, 0) A = ( B stationary: (x,t) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.18 SR: tachyonic antitelephone t ′ t ) ′C = ( L,t tC C = (αtC,tC) tachyon 0) x′ (L, ) B x L/α L, − (0, 0) A = ( B stationary: (x,t) frame A moving at speed β: (x′,t′) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.18 SR: tachyonic antitelephone t ′ t A: tachyon at α > 1 to B ) ′C = ( L,t tC C = (αtC,tC) tachyon 0) x′ (L, ) B x L/α L, − (0, 0) A = ( B stationary: (x,t) frame A moving at speed β: (x′,t′) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.18 SR: tachyonic antitelephone t ′ t B: tachyon at α > 1 to C ) ′C = ( L,t tC C = (αtC,tC) tachyon 0) x′ (L, ) B x L/α L, − (0, 0) A = ( B stationary: (x,t) frame A moving at speed β: (x′,t′) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.18 SR: tachyonic antitelephone t ′ t L αtC C: ′ = Λ tC tC ) ′C = ( L,t tC C = (αtC,tC) tachyon 0) x′ (L, ) B x L/α L, − (0, 0) A = ( B stationary: (x,t) frame A moving at speed β: (x′,t′) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.18 SR: tachyonic antitelephone t ′ t ) ′C = ( L,t tC C = (αt ,t ) L α β C C ′ = γtC − tC αβ + 1 tachyon 0) − x′ (L, ) B x L/α L, − (0, 0) A = ( B stationary: (x,t) frame A moving at speed β: (x′,t′) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.18 SR: tachyonic antitelephone t ′ t ) L α β L,t′C ′ = γtC − t = ( C C = (αtC,tC) tC αβ + 1 ′ − tC = γtC(1 αβ) tachyon 0) x′ (L, − ) B x L/α L, − (0, 0) A = ( B stationary: (x,t) frame A moving at speed β: (x′,t′) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.18 SR: tachyonic antitelephone t ′ t L α β ) = γtC − L,t′C ′ t = ( tC αβ + 1 C C = (αtC,tC) − ′ L tC = γ γ(α−β)(1 αβ) tachyon 0) − x′ (L, ) B x L/α L, − (0, 0) A = ( B stationary: (x,t) frame A moving at speed β: (x′,t′) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.18 SR: tachyonic antitelephone t ′ t ) ′C = ( L,t tC C = (αt ,t ) L α β C C ′ = γtC − tC αβ + 1 tachyon 0) − x′ (L, ′ 1−αβ ) tC = L α−β B x L/α L, − (0, 0) A = ( B stationary: (x,t) frame A moving at speed β: (x′,t′) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.18 SR: tachyonic antitelephone t ′ t L α β = γt − ) ′ C L,t′C tC αβ + 1 t = ( − C C = (αtC,tC) ′ 1−αβ tC = L α−β tachyon 0) x′ (L, ′ ′ 1−αβ 1 t t = L − + ) C A α β α B x L/α − L, − (0, 0) A = ( B stationary: (x,t) frame A moving at speed β: (x′,t′) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.18 SR: tachyonic antitelephone t ′ t L α β ) ′ = γtC − ′C = ( L,t tC αβ + 1 tC C = (αt ,t ) − C C ′ 1−αβ tC = L α−β tachyonx′ 0) − 2 − (L, t′ t′ = L α α β+α β ) C A α(α−β) B x L/α − L, − (0, 0) A = ( B stationary: (x,t) frame A moving at speed β: (x′,t′) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.18 SR: tachyonic antitelephone t ′ t ) ′C = ( L,t tC C = (αt ,t ) L α β C C ′ = γtC − tC αβ + 1 tachyon 0) − x′ (L, ′ 1−αβ tC = L − B x ) α β L/α − 2 L, − ′ ′ 2α (α +1)β (0, 0) t t = L A = ( C − A α(α−β) B stationary: (x,t) frame A moving at speed β: (x′,t′) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.18 SR: tachyonic antitelephone t ′ t ) ′C = ( L,t tC C = (αt ,t ) L α β C C ′ = γtC − tC αβ + 1 tachyon 0) − x′ (L, ′ 1−αβ tC = L − B x ) α β L/α − 2 L, − ′ ′ 2α (α +1)β (0, 0) t t = L A = ( C − A α(α−β) 2α < 0 if β > α2+1 B stationary: (x,t) frame A moving at speed β: (x′,t′) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.18 SR: tachyonic antitelephone t ′ t ) ′C = ( L,t tC C = (αt ,t ) L α β C C ′ = γtC − tC αβ + 1 tachyon 0) − x′ (L, ′ 1−αβ tC = L − B x ) α β L/α − 2 L, − ′ ′ 2α (α +1)β (0, 0) t t = L A = ( C − A α(α−β) 2α < 0 if β > α2+1 B stationary: (x,t) A receives tachyonic re- frame sponse at C before sending A moving at speed β: it (x′,t′) frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.18 SR: tachyonic antitelephone t ′ t ) ′C = ( L,t tC C = (αt ,t ) L α β C C ′ = γtC − tC αβ + 1 tachyon 0) − x′ (L, ′ 1−αβ tC = L − B x ) α β L/α − 2 L, − ′ ′ 2α (α +1)β (0, 0) t t = L A = ( C − A α(α−β) 2α < 0 if β > α2+1 B stationary: (x,t) A receives tachyonic re- frame sponse at C before sending A moving at speed β: it (x′,t′) frame w:tachyonic antitelephone 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.18 SR: pole-barn/ladder paradox 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.19 SR: pole-barn/ladder paradox ladder of length 29.9γ ns, garage length 30 ns • 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.19 SR: pole-barn/ladder paradox ladder of length 29.9γ ns, garage length 30 ns • instantaneously close front + back doors • 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.19 SR: pole-barn/ladder paradox ladder of length 29.9γ ns, garage length 30 ns • instantaneously close front + back doors • 29.9γ ns /γ < 30 ns OK • ⇒ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.19 SR: pole-barn/ladder paradox ladder of length 29.9γ ns, garage length 30 ns • instantaneously close front + back doors • ladder frame: garage 30/γ ns long 29.9γ ns!! • ≪ Is this possible or not? Make a spacetime diagram. 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.19 SR: pole-barn/ladder paradox t t ′ x′ γ ns 29.9 x (0,0) (30 ns,0) ns 30/γ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.19 SR: pole-barn/ladder paradox 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.19 SR: pole-barn/ladder paradox 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.19 SR: pole-barn/ladder paradox w:Ladder paradox 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.19 SR: twins paradox t′ t D B2 B1 (0, L/β) (L, L/β) B3 p q worldline x′ A1 C x (0, 0) A2 =(L, 0) simply connected Minkowski 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.20 SR: twins paradox t′ t D B2 B1 (0, L/β) (L, L/β) B3 p ) p ( q g worldline x′ A1 C x (0, 0) A2 =(L, 0) holonomy g identify spacetime events 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.20 SR: twins paradox t t′ L B2 = 0, βγ D B1 B3 q p ) p ( g C x′ A1 = (0, 0) x A = (Lγ, βγL) 2 − g 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.20 SR: twins paradox B2 B1 p worldline A2 A1 x 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.20 SR: twins paradox D B3 B2 ) p ( g q C A2 A1 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.20 SR: twins paradox t t′ space−limited FD time−limited E3 FD C E 2 2 C1 E1 x′ A1 A2 x g 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.20 SR: twins paradox t′ q g(p) g(g(p)) C1 E1 A1 q C2 g(p) E2 g(g(p)) q x′ Roukema & Bajtlik 2008, MNRAS, 390, 655 arXiv:astro-ph/0612155 helps understand w:Ehrenfest paradox • 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.20 SR: four-velocity, four-momentum T choose x axis so that 3-velocity uGalilean =(β, 0, 0) for observer with (t,x,y,z)T coord system 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.21 SR: four-velocity, four-momentum T choose x axis so that 3-velocity uGalilean =(β, 0, 0) for observer with (t,x,y,z) coord system t t′ x′ tangent to worldline O x (0,0) in (t,x) spacetime •2-plane, extend from scalar speed β to spacetime vector = 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.21 tangent to worldline SR: four-velocity, four-momentum T choose x axis so that 3-velocity uGalilean =(β, 0, 0) for observer with (t,x,y,z) coord system t x d d (u , u ) := ( dτ t(τ), dτ x(τ)) t w:four-velocity t′ x′ tangent to worldline O x (0,0) in (t,x) spacetime •2-plane, extend from scalar speed β to spacetime vector = 0 M Λ tangentβ cal c Σφ γ to t+ worldline x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.21 SR: four-velocity, four-momentum T choose x axis so that 3-velocity uGalilean =(β, 0, 0) for observer with (t,x,y,z) coord system t x d d (u , u ) := ( dτ t(τ), dτ x(τ)) w:four-velocity ′ ′ similarly (ut , ux ) = t ( d t′(τ), d x′(τ)) t′ dτ dτ x′ tangent to worldline O x (0,0) in (t,x) spacetime • 0 M Λ 2-plane,β cal c Σφ γ extendt+ x− z xy from tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.21 scalar speed to SR: four-velocity, four-momentum T choose x axis so that 3-velocity uGalilean =(β, 0, 0) for observer with (t,x,y,z) coord system t x d d t (u , u ) := ( t(τ), x(τ)) t dτ dτ ′ w:four-velocity ′ ′ t x x′ similarly (u , u ) = ( d t′(τ), d x′(τ)) = (1, 0) tangent to worldline dτ dτ O x (0,0) in (t,x) spacetime •2-plane, extend from scalar speed β to spacetime vector = tangent to worldline 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.21 SR: four-velocity, four-momentum T choose x axis so that 3-velocity uGalilean =(β, 0, 0) for observer with (t,x,y,z) coord system t x d d (u , u ) := ( dτ t(τ), dτ x(τ)) w:four-velocity ′ ′ t t x t′ similarly (u , u ) = d ′ d ′ ( dτ t (τ), dτ x (τ)) = (1, 0) x′ want u Lorentz invariant (ut, ux)T = Λ−1(1, 0)T ⇒ tangent to worldline O x (0,0) in (t,x) spacetime •2-plane, extend from 0 M Λ scalarβ cal c Σφ speedγ t+ x− z βxy tc toPen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.21 spacetime vector = SR: four-velocity, four-momentum T choose x axis so that 3-velocity uGalilean =(β, 0, 0) for observer with (t,x,y,z) coord system t x d d t (u , u ) := ( t(τ), x(τ)) t dτ dτ ′ w:four-velocity ′ ′ t x x′ similarly (u , u ) = ( d t′(τ), d x′(τ)) = (1, 0) tangent to worldline dτ dτ O want u Lorentz invariant x t x T −1 T (0,0) (u , u ) = Λ (1, 0) = ⇒ γ(1, β)T in (t,x) spacetime •2-plane, extend from scalar speed β to spacetime vector = tangent to worldline 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.21 SR: four-velocity, four-momentum T choose x axis so that 3-velocity uGalilean =(β, 0, 0) for observer with (t,x,y,z) coord system t x d d t (u , u ) := ( t(τ), x(τ)) t dτ dτ ′ w:four-velocity ′ ′ t x x′ similarly (u , u ) = ( d t′(τ), d x′(τ)) = (1, 0) tangent to worldline dτ dτ O want u Lorentz invariant x t x T −1 T (0,0) (u , u ) = Λ (1, 0) = ⇒ γ(1, β)T in (t,x) spacetime 4D: u = γ(1, βx, βy, βz)T •2-plane, extend from notation in this pdf: scalar speed β to spacetime vector = u = 4-vector, (3) u = spatial tangent to worldline component 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.21 SR: four-velocity, four-momentum Is the (3)-component (spatial component) of u the same as the non-relativistic velocity? 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.22 SR: four-velocity, four-momentum Is the (3)-component (spatial component) of u the same as the non-relativistic velocity? (3) d T u = dτ (x,y,z) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.22 SR: four-velocity, four-momentum Is the (3)-component (spatial component) of u the same as the non-relativistic velocity? (3) d T u = dτ (x,y,z) d T = γ dt (x,y,z) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.22 SR: four-velocity, four-momentum Is the (3)-component (spatial component) of u the same as the non-relativistic velocity? (3) d T u = dτ (x,y,z) d T = γ dt (x,y,z) = d (x,y,z)T except if β = 0 γ = 1 dt ⇔ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.22 SR: four-velocity, four-momentum momentum: p := m u = mγ(1, βx, βy, βz)T, where m = constant w:invariant mass x ...= tensor-style component notation, not powers 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.23 SR: four-velocity, four-momentum momentum: p := m u = mγ(1, βx, βy, βz)T, where m = constant w:invariant mass 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.23 SR: four-velocity, four-momentum momentum: p := m u = mγ(1, βx, βy, βz)T, where m = constant w:invariant mass What does the time component of momentum = p0 = mγ mean physically? first look at spatial component in a given ref. frame • 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.23 SR: four-velocity, four-momentum momentum: p := m u = mγ(1, βx, βy, βz)T, where m = constant w:invariant mass (3) d T p = m dτ (x,y,z) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.23 SR: four-velocity, four-momentum momentum: p := m u = mγ(1, βx, βy, βz)T, where m = constant w:invariant mass (3) d T p = m dτ (x,y,z) d T = mγ dt (x,y,z) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.23 SR: four-velocity, four-momentum momentum: p := m u = mγ(1, βx, βy, βz)T, where m = constant w:invariant mass (3) d T p = m dτ (x,y,z) d T = mγ dt (x,y,z) = m d (x,y,z)T except if β = 0 γ = 1 dt ⇔ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.23 SR: four-velocity, four-momentum let us define 4-acceleration, 4-force 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.24 SR: four-velocity, four-momentum t x d d (u , u ):=( dτ t(τ), dτ x(τ)) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.24 SR: four-velocity, four-momentum d a := dτ u 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.24 SR: four-velocity, four-momentum d a := dτ u (3) d2 (3) a = dτ 2 x 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.24 SR: four-velocity, four-momentum d a := dτ u (3) d2 T a = dτ 2 (x,y,z) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.24 SR: four-velocity, four-momentum d a := dτ u (4)f := m (4) a defn w:four-force 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.24 SR: four-velocity, four-momentum d a := dτ u (4)f := m (4) a defn w:four-force d = m dτ u 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.24 SR: four-velocity, four-momentum d a := dτ u (4)f := m (4) a defn w:four-force d = dτ p 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.24 SR: invariance of (4)u, (4)a, (4)f Euclidean norm: x 2 = (x )2 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.25 SR: invariance of (4)u, (4)a, (4)f Minkowski pseudo-norm: x 2 = η x xν ,ν ν 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.25 SR: invariance of (4)u, (4)a, (4)f Minkowski pseudo-norm: x 2 = η x xν ν w:Einstein summation sum is implicit 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.25 SR: invariance of (4)u, (4)a, (4)f Minkowski pseudo-norm: x 2 = x0x0 + δ xixj − ij δ = 1 if i = j, otherwise = 0; i, j 1, 2, 3 ij ∈ invariance: x 2 = same in all reference frames sign convention: ( , +, +, +) or (+, , , ) are common − − − − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.25 SR: invariance of (4)u, (4)a, (4)f Minkowski pseudo-norm: x 2 = x0x0 + δ xixj − ij δ = 1 if i = j, otherwise = 0; i, j 1, 2, 3 ij ∈ invariance: x 2 = same in all reference frames sign convention: ( , +, +, +) or (+, , , ) are common − − − − non-rest frame: u 2 = γ2 + γ2β2 − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.25 SR: invariance of (4)u, (4)a, (4)f Minkowski pseudo-norm: x 2 = x0x0 + δ xixj − ij δ = 1 if i = j, otherwise = 0; i, j 1, 2, 3 ij ∈ invariance: x 2 = same in all reference frames sign convention: ( , +, +, +) or (+, , , ) are common − − − − non-rest frame: u 2 = γ2 + γ2β2 = 1 − − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.25 SR: invariance of (4)u, (4)a, (4)f Minkowski pseudo-norm: x 2 = x0x0 + δ xixj − ij δ = 1 if i = j, otherwise = 0; i, j 1, 2, 3 ij ∈ invariance: x 2 = same in all reference frames sign convention: ( , +, +, +) or (+, , , ) are common − − − − non-rest frame: u 2 = γ2 + γ2β2 = 1 − − rest frame: u 2 = 12 + 02 = 1 invariant − − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.25 SR: invariance of (4)u, (4)a, (4)f Minkowski pseudo-norm: x 2 = x0x0 + δ xixj − ij δ = 1 if i = j, otherwise = 0; i, j 1, 2, 3 ij ∈ invariance: x 2 = same in all reference frames sign convention: ( , +, +, +) or (+, , , ) are common − − − − non-rest frame: u 2 = γ2 + γ2β2 = 1 − − rest frame: u 2 = 12 + 02 = 1 invariant − − similarly: a 2, f 2 invariant 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.25 SR: energy: varies with ref frame Newtonian K = (1/2)mβ2 = 0 in rest frame 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.26 SR: energy: varies with ref frame Newtonian K = (1/2)mβ2 = 0 in rest frame 4-force f is invariant, but 3-force usually defined to be frame-dependent: 3-force := d (3) p = d (3) p dt dτ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.26 SR: energy: varies with ref frame Newtonian K = (1/2)mβ2 = 0 in rest frame 4-force f is invariant, but 3-force usually defined to be frame-dependent: 3-force := d (3) p = d (3) p dt dτ d (3) (3)f dt p = γ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.26 SR: energy: varies with ref frame in (x,t) frame, K = work done 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.27 SR: energy: varies with ref frame β2 (3)f in (x,t) frame, = d x γ K = work done 0 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.27 SR: energy: varies with ref frame β2 (3)f in (x,t) frame, = d x 0 γ K = work done β 2 d = (3) p d x dt 0 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.27 SR: energy: varies with ref frame β2 (3)f in (x,t) frame, = d x 0 γ K = work done β 2 d = (mβγ)dx dt 0 (assume (3)f/γ x ) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.27 SR: energy: varies with ref frame β2 (3)f in (x,t) frame, = d x 0 γ K = work done β β 2 d 2 dx = (mβγ)dx = m d(βγ) dt dt 0 0 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.27 SR: energy: varies with ref frame β2 (3)f in (x,t) frame, = d x 0 γ K = work done β β 2 d 2 = (mβγ)dx = m d(βγ)β dt 0 0 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.27 SR: energy: varies with ref frame β2 (3)f in (x,t) frame, = d x 0 γ K = work done β β 2 d 2 = (mβγ)dx = m d(βγ)β 0 dt 0 β2 = m β(βdγ + γdβ) 0 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.27 SR: energy: varies with ref frame β2 (3)f in (x,t) frame, = d x 0 γ K = work done β β 2 d 2 = (mβγ)dx = m d(βγ)β 0 dt 0 β2 = m β(βdγ + γdβ) 0 γ=γ2 − = m (β2 + γ 2)dγ dγ = βγ3dβ ⇐ γ=1 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.27 SR: energy: varies with ref frame β2 (3)f in (x,t) frame, = d x 0 γ K = work done β β 2 d 2 = (mβγ)dx = m d(βγ)β 0 dt 0 β2 = m β(βdγ + γdβ) 0 γ=γ2 − = m (β2 + γ 2)dγ dγ = βγ3dβ γ=1 ⇐ γ=γ2 = m [β2 + (1 β2)]dγ − γ=1 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.27 SR: energy: varies with ref frame β2 (3)f in (x,t) frame, = d x 0 γ K = work done β β 2 d 2 = (mβγ)dx = m d(βγ)β 0 dt 0 β2 = m β(βdγ + γdβ) 0 γ=γ2 − = m (β2 + γ 2)dγ dγ = βγ3dβ γ=1 ⇐ γ=γ2 = m dγ γ=1 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.27 SR: energy: varies with ref frame β2 (3)f in (x,t) frame, = d x 0 γ K = work done β β 2 d 2 = (mβγ)dx = m d(βγ)β 0 dt 0 β2 = m β(βdγ + γdβ) 0 γ=γ2 − = m (β2 + γ 2)dγ dγ = βγ3dβ γ=1 ⇐ γ=γ2 = m dγ = mγ m 2 − γ=1 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.27 SR: energy: varies with ref frame β2 (3)f in (x,t) frame, = d x 0 γ K = work done β β 2 d 2 = (mβγ)dx = m d(βγ)β 0 dt 0 β2 = m β(βdγ + γdβ) 0 γ=γ2 − = m (β2 + γ 2)dγ dγ = βγ3dβ γ=1 ⇐ γ=γ2 = m dγ = mγ m 2 − γ=1 K + m = mγ drop “ " ⇒ 2 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.27 SR: energy: varies with ref frame β2 (3)f in (x,t) frame, = d x 0 γ K = work done β β 2 d 2 = (mβγ)dx = m d(βγ)β 0 dt 0 β2 = m β(βdγ + γdβ) 0 γ=γ2 − = m (β2 + γ 2)dγ dγ = βγ3dβ γ=1 ⇐ γ=γ2 = m dγ = mγ m 2 − γ=1 K + m = mγ = p0 ⇒ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.27 SR: energy: varies with ref frame β2 (3)f in (x,t) frame, = d x 0 γ K = work done β β 2 d 2 = (mβγ)dx = m d(βγ)β 0 dt 0 β2 = m β(βdγ + γdβ) 0 γ=γ2 − = m (β2 + γ 2)dγ dγ = βγ3dβ γ=1 ⇐ γ=γ2 = m dγ = mγ m 2 − γ=1 K + m = mγ = p0 ⇒ so p0 = kinetic energy + rest mass 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.27 SR: energy: varies with ref frame Does small β limit agree with Newtonian K? 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.28 SR: energy: varies with ref frame Does small β limit agree with Newtonian K? momentum time component: p0 = mγ = m(1 β2)−1/2 − = m[1 (1/2)( β2) + (β4)] if β 1 − − O ≪ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.28 SR: energy: varies with ref frame Does small β limit agree with Newtonian K? momentum time component: p0 = mγ = m(1 β2)−1/2 − m[1+(1/2)β2] if β 1 ≈ ≪ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.28 SR: energy: varies with ref frame Does small β limit agree with Newtonian K? momentum time component: p0 = mγ = m(1 β2)−1/2 − m + (1/2)mβ2 if β 1 ≈ ≪ Yes. 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.28 SR: p... : invariant or not? momentum: p = mγ(1, βx, βy, βz)T p0 = m + K = mγ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.29 SR: p... : invariant or not? momentum: p = mγ(1, βx, βy, βz)T p0 = m + K = mγ non-rest frame: u 2 = γ2 + γ2β2 = 1 − − rest frame: u 2 = 12 + 02 = 1 invariant − − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.29 SR: p... : invariant or not? momentum: p = mγ(1, βx, βy, βz)T p0 = m + K = mγ non-rest frame: p 2 = m2γ2 + m2γ2β2 = m2 − − rest frame: p 2 = m2 + 02 = m2 invariant − − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.29 SR: p... : invariant or not? momentum: p = mγ(1, βx, βy, βz)T p0 = m + K = mγ non-rest frame: p 2 = m2γ2 + m2γ2β2 = m2 − − rest frame: p 2 = m2 + 02 = m2 invariant − − m = w:invariant mass rest mass: invariant ≡ AND conserved (in interactions): p + q 2 = r 2 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.29 SR: p... : invariant or not? momentum: p = mγ(1, βx, βy, βz)T p0 = m + K = mγ non-rest frame: p 2 = m2γ2 + m2γ2β2 = m2 − − rest frame: p 2 = m2 + 02 = m2 invariant − − m = w:invariant mass rest mass: invariant ≡ AND conserved (in interactions): p + q 2 = r 2 where interaction is (4-momenta): p + q r → 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.29 SR: p... : invariant or not? momentum: p = mγ(1, βx, βy, βz)T p0 = m + K = mγ non-rest frame: p 2 = m2γ2 + m2γ2β2 = m2 − − rest frame: p 2 = m2 + 02 = m2 invariant − − m = w:invariant mass rest mass: invariant ≡ AND conserved (in interactions): p + q 2 = r 2 where interaction is (4-momenta): p + q r → WARNING: assume that 4-momentum vectors at different space-time positions can be parallel-transported; not the case in curved spacetime 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.29 SR: p... : invariant or not? vector space pi + qi = ri (i = 1, 2, 3) ⇒ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.30 SR: p... : invariant or not? vector space pi + qi = ri (i = 1, 2, 3) ⇒ = conservation of (relativistic) (3)-momentum (Newtonian: conserved) but NOT invariant (Newtonian: not invariant) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.30 SR: p... : invariant or not? vector space pi + qi = ri (i = 1, 2, 3) ⇒ = conservation of (relativistic) (3)-momentum (Newtonian: conserved) but NOT invariant (Newtonian: not invariant) vector space p0 + q0 = r0 ⇒ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.30 SR: p... : invariant or not? vector space pi + qi = ri (i = 1, 2, 3) ⇒ = conservation of (relativistic) (3)-momentum (Newtonian: conserved) but NOT invariant (Newtonian: not invariant) vector space p0 + q0 = r0 ⇒ = conservation of (relativistic) “total energy" = m + K (Newtonian: m conserved, K not conserved, K+ potential energy conserved) but NOT invariant (Newtonian: m invariant, K not invariant) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.30 SR: p... : invariant or not? t t t′′ ′ =? 2 q =0 + x p x′ x q x 2 t 2 ′′ = = m − x − − x β m = β 2 2 = p t 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.31 SR: p... : invariant or not? t t t′′ ′ =? 2 q =0 + x p x′ x q x 2 t 2 ′′ = = m − x − − x β m = β 2 2 = p t p = m(γ, +βγ, 0, 0)T 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.31 SR: p... : invariant or not? t t t′′ ′ =? 2 q =0 + x p x′ x q x 2 t 2 ′′ = = m − x − − x β m = β 2 2 = p t p = m(γ, +βγ, 0, 0)T q = m(γ, βγ, 0, 0)T − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.31 SR: p... : invariant or not? t t t′′ ′ =? 2 q =0 + x p x′ x q x 2 t 2 ′′ = = m − x − − x β m = β 2 2 = p t p = m(γ, +βγ, 0, 0)T q = m(γ, βγ, 0, 0)T r = p + q− 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.31 SR: p... : invariant or not? t t t′′ ′ =? 2 q =0 + x p x′ x q x 2 t 2 ′′ = = m − x − − x β m = β 2 2 = p t p = m(γ, +βγ, 0, 0)T q = m(γ, βγ, 0, 0)T r = p + q− = m[2γ, ( β + β)γ, 0, 0]T − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.31 SR: p... : invariant or not? t t t′′ ′ =? 2 q =0 + x p x′ x q x 2 t 2 ′′ = = m − x − − x β m = β 2 2 = p t p = m(γ, +βγ, 0, 0)T q = m(γ, βγ, 0, 0)T − r = p+ q = 2mγ(1, 0, 0, 0)T 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.31 SR: p... : invariant or not? t t t′′ ′ system rest mass be- fore and after: 2mγ =? 2 q =0 + x p x′ x q x 2 t 2 ′′ = = m − x − − x β m = β 2 2 = p t p = m(γ, +βγ, 0, 0)T q = m(γ, βγ, 0, 0)T − r = p+ q = 2mγ(1, 0, 0, 0)T 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.31 SR: p... : invariant or not? t t t′′ ′ system rest mass be- fore and after: 2mγ =? 2 rest masses in many q =0 + x p different frames: x ′ m + m = 2mγ if γ = 1 x q x 2 t 2 ′′ = = m − x − − x β m = β 2 2 = p t p = m(γ, +βγ, 0, 0)T q = m(γ, βγ, 0, 0)T − r = p+ q = 2mγ(1, 0, 0, 0)T 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.31 SR: p... : invariant or not? t t t′′ ′ system rest mass be- fore and after: 2mγ =? 2 rest masses in many q =0 + x p different frames: x ′ m + m = 2mγ if γ = 1 x p 2 + q 2 = q − − 2 x′′ 2 t 2 = = m r − x − − x β m − = β 2 2 = p t p = m(γ, +βγ, 0, 0)T q = m(γ, βγ, 0, 0)T − r = p+ q = 2mγ(1, 0, 0, 0)T 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.31 SR: p... : invariant or not? t t t′′ ′ system rest mass be- fore and after: 2mγ =? 2 rest masses in many q =0 + x p different frames: x ′ m + m = 2mγ if γ = 1 x p 2 + q 2 = q − − 2 x′′ 2 t 2 = = m r − x − − x β m − = β 2 2 = system mass is invari- p t ant, but can be di- T p = m(γ, +βγ, 0, 0) vided into p0 and pi, i q = m(γ, βγ, 0, 0)T 1, 2, 3 components in∈ − { } r = p+ q = 2mγ(1, 0, 0, 0)T many different ways 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.31 SR: p... : invariant or not? interaction: momenta: p + q r → moduli: m, m, 2mγ 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.32 SR: p... : invariant or not? interaction: momenta: p + q r → moduli: m, m, 2mγ total energy E := p0 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.32 SR: p... : invariant or not? interaction: momenta: p + q r → moduli: m, m, 2mγ total energy E := p0 rest frame: E := p0 = t component of m(1, 0, 0, 0)T E = m (the famous equation) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.32 SR: p... : invariant or not? interaction: momenta: p + q r → moduli: m, m, 2mγ total energy E := p0 rest frame: E := p0 = t component of m(1, 0, 0, 0)T E = m (the famous equation) this means: in the rest frame, K + m = m (trivial) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.32 SR: p... : invariant or not? interaction: momenta: p + q r → moduli: m, m, 2mγ total energy E := p0 rest frame: E := p0 = t component of m(1, 0, 0, 0)T E = m (the famous equation) this means: in the rest frame, K + m = m (trivial) more interesting: non-rest frame: 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.32 SR: p... : invariant or not? interaction: momenta: p + q r → moduli: m, m, 2mγ total energy E := p0 rest frame: E := p0 = t component of m(1, 0, 0, 0)T E = m (the famous equation) this means: in the rest frame, K + m = m (trivial) more interesting: non-rest frame: m2 = p 2 − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.32 SR: p... : invariant or not? interaction: momenta: p + q r → moduli: m, m, 2mγ total energy E := p0 rest frame: E := p0 = t component of m(1, 0, 0, 0)T E = m (the famous equation) this means: in the rest frame, K + m = m (trivial) more interesting: non-rest frame: m2 = p 2 − m2 = (E,px,py,pz)T 2 − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.32 SR: p... : invariant or not? interaction: momenta: p + q r → moduli: m, m, 2mγ total energy E := p0 rest frame: E := p0 = t component of m(1, 0, 0, 0)T E = m (the famous equation) this means: in the rest frame, K + m = m (trivial) more interesting: non-rest frame: m2 = p 2 − m2 = (E,px,py,pz)T 2 − m2 = E2 + p2 − − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.32 SR: p... : invariant or not? interaction: momenta: p + q r → moduli: m, m, 2mγ total energy E := p0 rest frame: E := p0 = t component of m(1, 0, 0, 0)T E = m (the famous equation) this means: in the rest frame, K + m = m (trivial) more interesting: non-rest frame: m2 = p 2 − m2 = (E,px,py,pz)T 2 − m2 = E2 p2 − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.32 SR: p... : invariant or not? interaction: momenta: p + q r → moduli: m, m, 2mγ total energy E := p0 rest frame: E := p0 = t component of m(1, 0, 0, 0)T E = m (the famous equation) this means: in the rest frame, K + m = m (trivial) more interesting: non-rest frame: m2 = p 2 − m2 = (E,px,py,pz)T 2 − m2 = E2 (3) p 2 − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.32 SR: null 4-momentum photon: m = 0 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.33 SR: null 4-momentum photon: m = 0 extend defn of 4-momentum to p for a photon 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.33 SR: null 4-momentum photon: m = 0 extend defn of 4-momentum to p for a photon u = p since m = 0; no 4-velocity ⇒ m 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.33 SR: null 4-momentum photon: m = 0 extend defn of 4-momentum to p for a photon u = p since m = 0; no 4-velocity ⇒ m so 0 = E2 p2 − 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.33 SR: null 4-momentum photon: m = 0 extend defn of 4-momentum to p for a photon u = p since m = 0; no 4-velocity ⇒ m so 0 = E2 p2 − i.e. p =(E,E, 0, 0)T =(p,p, 0, 0)T (if x direction) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.33 SR: null 4-momentum photon: m = 0 extend defn of 4-momentum to p for a photon u = p since m = 0; no 4-velocity ⇒ m so 0 = E2 p2 − i.e. p =(E,E, 0, 0)T =(p,p, 0, 0)T (if x direction) so E = p for any massless particle 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.33 SR: model summary Minkowski spacetime: draw a correct diagram 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.34 SR: model summary Minkowski spacetime: draw a correct diagram Lorentz transformation (boost) Λ(φ) or Λ(β) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR ▽ – p.34 SR: model summary Minkowski spacetime: draw a correct diagram Lorentz transformation (boost) Λ(φ) or Λ(β) refuse the assumption of absolute simultaneity (time) 0 M Λ β cal c Σφ γ t+ x− z xy tc Pen β+ PB tw u p p E=m E=p SR SR+ǫGR – p.34