a<Time Dilation> c𝛥t=2hc𝛥t'=2(v⋅𝛥t'

2)2+h2=(v𝛥t')2+4h2= (v𝛥t')2+(2h)2(c𝛥t')2=(v𝛥t')2+(c𝛥t)2c2(𝛥t)2=(c2-v2)(𝛥t')2(𝛥t)2=(1-(v

c)2)(𝛥t')2𝛥t= 1-(v

c)2𝛥t' 𝛾≡1

1-(v

c)2 𝛥t'=𝛾 𝛥t <𝛽 and 𝛾>0⩽v<c0⩽v

c<1𝛽≡v

c0⩽𝛽<1 0⩽𝛽2<1-1<-𝛽2⩽00<1-𝛽2⩽10<1-𝛽2⩽1𝛾=1

1-𝛽2=(1-𝛽2)-1/21

𝛾⩽1𝛾⩾1 <Length Contraction>𝛥x'=v𝛥t𝛥x=v𝛥t'=𝛾 v𝛥t=𝛾 𝛥x'𝛥x'=𝛥x

𝛾 <Lorentz Transformation>An obeserver and a traveller meets at some point in spacetime. And they agree to define each one's reference frame with that point in spacetime as origin.So the (0,0,0,0) point is the same point in spacetime for both of them.The traveller is travelling with relative speed of v in some direction as seen by the observer.The observer defines x axis as the direction the traveller is heading.From the traveller's point of view, the observer is travelling in the opposite direction with relative speed of v.Nevertheless, the traveller agrees to define x' axis in the same direction as the observer.They also agree on the directions of y and y' axis, z and z' axis.Lastly, they agree to call the observer's reference frame as S, and the traveller's reference frame as S'. S:(t,x,y,z)S':(t',x',y',z') 1 [m] = (1 [m/s] )(1 [s])1 [m] = (c [m/s] )(1/c [s])c [m] = (c [m/s] )(1 [s]) S:(t[s],x[m],y[m],z[m])S':(t'[s],x'[m],y'[m],z'[m]) They want to represent a point(or an event) in spacetime in their reference frames as a vector.But the unit of time is different from the unit of distance.They can't just add 2 vectors like they would with position vectors.So they came up with a solution.Since the speed of light in vaccum doesn't change, by multiplying time with the speed of light,the unit for representing time becomes the unit of distance. Sc:(ct[m],x[m],y[m],z[m])S'c:(ct'[m],x'[m],y'[m],z'[m]) They now call these modified reference frames Sc and S'c. The observer wants to know how ct' axis can be represented in Sc.Along the ct' axis, x',y',z' values must be 0.At the traveller's position, x'=y'=z'=0 is always satisfied. Since the traveller moves only in x direction,ct' axis unit vector, ct' must have no y and z component.Because of time dilation, 1 second for the traveller is 𝛾 seconds for the observer.Therefore, the traveller travels v𝛾 meters in Sc for 1 second in S'c. ct'=(c[m/s]⋅𝛾[s],v[m/s]⋅𝛾[s],0[m],0[m])ct'=(c𝛾,v𝛾,0,0) Along the x' axis, t',y',z' values must be 0.t'=0 means that events on the x' axis are simultaneous for the traveller.The traveller has a light source and 2 mirrors.Those mirrors are both 1 meter away from the traveller, measured by the traveller.Specifically, y'=z'=0 for both, and one at x'=1, and the other at x'=-1. The traveller turns on the light source at t'=0. At t'=(1/c),pulses of light emitted from the light source reach both mirrors 'Simultaneously' from the perspective of the traveller. At t'=(2/c),the traveller sees pulses of light from both directionsarriving simultaneously. On the other hand, from the perspective of the observer,due to time dilation, (1/c) seconds for the traveller is (𝛾/c) seconds.During this time, light travels c⋅(𝛾/c)=𝛾 meters, and the mirrors travel v⋅(𝛾/c)=(𝛾v/c) meters.Therefore, light headed in  x direction has to travel 𝛾(1+(v/c)) meters to reach the mirror,and light headed in -x direction has to travel 𝛾(1-(v/c)) meters to reach the other mirror.Which means, light headed in  x direction reaches the mirror at t=(𝛾/c)(1+(v/c)),and light headed in -x direction reaches the other mirror at t=(𝛾/c)(1-(v/c)).Events on the line passing through these 2 events in spacetime are simultaneous.Thus, x' axis unit vector, x' can be obtained as below. x'=1

2a(c[m/s]⋅(𝛾⋅1[m]

c[m/s])(1+v[m/s]⋅1

c[m/s]),(c[m/s]+v[m/s])⋅(𝛾⋅1[m]

c[m/s]),0[m],0[m])-(c[m/s]⋅(𝛾⋅1[m]

c[m/s])(1-v[m/s]⋅1

c[m/s]),(-c[m/s]+v[m/s])⋅(𝛾⋅1[m]

c[m/s]),0[m],0[m])ax'=1

2{(𝛾+v

c𝛾,c+v

c𝛾,0,0)-(𝛾-v

c𝛾,v-c

c𝛾,0,0)}x'=1

2(2v

c𝛾,2c

c𝛾,0,0)x'=(v

c𝛾,𝛾,0,0) (ct,x,y,z)=t'ct'+x'x'+(0,0,y',z')(ct,x,y,z)=t'(c𝛾,v𝛾,0,0)+x'(v

c𝛾,𝛾,0,0)+(0,0,y',z')y=y'z=z'(ct,x,0,0)=(𝛾ct',𝛾vt',0,0)+(𝛾·v

cx',𝛾x',0,0)(ct,x,0,0)=(𝛾(ct'+v

cx'),𝛾(vt'+x'),0,0)(1)ct=𝛾(ct'+v

cx')(2)x=𝛾(vt'+x') (3)vt=𝛾(vt'+(v

c)2x')(4)v

cx=𝛾(v2

ct'+v

cx') (1)-(4)ct-v

cx=𝛾(c-v2

c)t't-vx

c2=𝛾(1-(v

c)2)t't-vx

c2=𝛾⋅𝛾-2t'𝛾-1t'=t-vx

c2(a)t'=𝛾(t-vx

c2)ct'=𝛾(ct-v

cx)𝛽=v

c(b)ct'=𝛾(ct-𝛽x) (2)-(3)x-vt=𝛾(1-(v

c)2)x'x-vt=𝛾⋅𝛾-2x'𝛾-1x'=x-vt(c)x'=𝛾(x-vt)(d)x'=𝛾(x-𝛽ct) -Lorentz boost- ct'=𝛾(ct-𝛽x)x'=𝛾(x-𝛽ct)y'=yz'=z -Inverse Lorentz boost- ct=𝛾(ct'+𝛽x')x=𝛾(x'+𝛽ct')y=y'z=z' <Lorentz Transformation of the differences>c𝛥t'=𝛾(c𝛥t-𝛽𝛥x)𝛥x'=𝛾(𝛥x-𝛽c𝛥t)𝛥y'=𝛥y𝛥z'=𝛥z c𝛥t=𝛾(c𝛥t'+𝛽𝛥x')𝛥x=𝛾(𝛥x'+𝛽c𝛥t')𝛥y=𝛥y'𝛥z=𝛥z' <Relativistic Velocity Addition>d

dt'(x',y',z')=(u'x,u'y,u'z)=u'd

dt'(𝛾(x-vt),y,z)=(u'x,u'y,u'z)d

dtdt

dt'(𝛾(x-vt),y,z)=(u'x,u'y,u'z) t=𝛾t'+𝛾v

c2x'dt

dt'=𝛾+𝛾v

c2u'x=𝛾(1+vu'x

c2) d

dt(𝛾2(1+vu'x

c2)(x-vt),𝛾(1+vu'x

c2)y,𝛾(1+vu'x

c2)z)=(u'x,u'y,u'z) d

dt(𝛾2(1+vu'x

c2)(x-vt))=u'x𝛾2(1+vu'x

c2)d

dt(x-vt)=u'x𝛾2(1+vu'x

c2)(dx

dt-v)=u'xdx

dt-v=u'x

𝛾2(1+vu'x

c2)dx

dt=u'x

c2+vu'x

c2

1-𝛽2+vdx

dt=u'x

c2+vu'x

c2-v2+vdx

dt=(c2-v2)u'x

c2+vu'x+vdx

dt=c2u'x-v2u'x+c2v+v2u'x

c2+vu'xdx

dt=c2(v+u'x)

c2+vu'xdx

dt=v+u'x

1+vu'x

c2 dy

dt=u'y

𝛾(1+vu'x

c2) , dz

dt=u'z

𝛾(1+vu'x

c2) d

dt(x,y,z)=(1

1+vu'x

c2)(v+u'x,u'y

𝛾,u'z

𝛾)=uu=(1

1+vu'x

c2)((v+u'x)x+1

𝛾(0,u'y,u'z))1

1+vu'x

c2≡𝛤'(0,u'y,u'z)≡u'yz u=𝛤'((v+u'x)x+u'yz

𝛾)u'=𝛤((-v+ux)x'+uyz

𝛾) 1

1-vux

c2≡𝛤(0,uy,uz)≡uyz <Minkowski Diagram>

ct'=(c𝛾,v𝛾,0,0)x'=(v

c𝛾,𝛾,0,0)v𝛾

c𝛾=v

c𝛾

𝛾=v

c=𝛽=tan𝛼 <Invariant interval> 𝛥s2≡c2𝛥t2-(𝛥x2+𝛥y2+𝛥z2) c𝛥t=𝛾(c𝛥t'+𝛽𝛥x')𝛥x=𝛾(𝛥x'+𝛽c𝛥t')𝛥y=𝛥y'𝛥z=𝛥z' 𝛥s2=𝛾2(c𝛥t'+𝛽𝛥x')2-(𝛾2(𝛥x'+𝛽c𝛥t')2+𝛥y'2+𝛥z'2)=𝛾2{(c𝛥t'+𝛽𝛥x')2-(𝛥x'+𝛽c𝛥t')2}-(𝛥y'2+𝛥z'2)=𝛾2{(c𝛥t'+𝛽𝛥x')+(𝛥x'+𝛽c𝛥t')}{(c𝛥t'+𝛽𝛥x')-(𝛥x'+𝛽c𝛥t')}-(𝛥y'2+𝛥z'2)=𝛾2{(1+𝛽)c𝛥t'+(1+𝛽)𝛥x'}{(1-𝛽)c𝛥t'-(1-𝛽)𝛥x'}-(𝛥y'2+𝛥z'2)=𝛾2(1-𝛽2)(c2𝛥t'2-𝛥x'2)-(𝛥y'2+𝛥z'2)=c2𝛥t'2-(𝛥x'2+𝛥y'2+𝛥z'2) c2𝛥t2-(𝛥x2+𝛥y2+𝛥z2)=𝛥s2=c2𝛥t'2-(𝛥x'2+𝛥y'2+𝛥z'2) <Rapidity> Hyperbolic number and functions
𝛽≡tanh𝜙 cosh2𝜙-sinh2𝜙=11-tanh2𝜙=1

cosh2𝜙 𝛾=1

1-𝛽2=1

1-tanh2𝜙=cosh𝜙 𝛾𝛽=𝛽

1-𝛽2=tanh𝜙

1-tanh2𝜙=sinh𝜙 ct'=𝛾(ct-𝛽x)x'=𝛾(x-𝛽ct)y'=yz'=z a

ct'

x'

=a

𝛾	-𝛾𝛽
-𝛾𝛽	𝛾

a

ct

x

a

ct'

x'

=a

cosh𝜙	-sinh𝜙
-sinh𝜙	cosh𝜙

a

ct

x

j2≡+1 (j∉ℜ) (ct'+x'j)=(ct+xj)⋅ej(-𝜙) ct=𝛾(ct'+𝛽x')x=𝛾(x'+𝛽ct')y=y'z=z' a

ct

x

=a

𝛾	𝛾𝛽
𝛾𝛽	𝛾

a

ct'

x'

a

ct

x

=a

cosh𝜙	sinh𝜙
sinh𝜙	cosh𝜙

a

ct'

x'

(ct+xj)=(ct'+x'j)⋅ej𝜙 u=𝛤'((v+u'x)x+u'yz

𝛾)ux=𝛤'(v+ux') 𝛽≡v

c≡tanh𝜙 , 𝛽u≡ux

c≡tanh𝜙u , 𝛽'u≡u'x

c≡tanh𝜙'u 𝛤'=1

1+vu'x

c2𝛤'=1

1+𝛽𝛽'u 𝛽u=𝛤'(𝛽+𝛽'u)𝛽u=𝛽+𝛽'u

1+𝛽𝛽'utanh𝜙u=tanh𝜙+tanh𝜙'u

1+tanh𝜙tanh𝜙'utanh𝜙u=tanh(𝜙+𝜙'u)𝜙u=𝜙+𝜙'u <Relativistic Doppler Effect> -in source's rest frame period isT0 -in observer's rest frame period isT=𝛾T0 in short timespan, only radial velocity changes distance

𝜃rrsv 𝜆'=cT+(vcos𝜃r)Tc=f'𝜆'c

f'=(c+vcos𝜃r)𝛾T0f'=c

(c+vcos𝜃r)𝛾T0f'=c

𝛾(c+vcos𝜃r)f0f'=f0

𝛾(1+𝛽cos𝜃r) <Relativistic Aberration> -in source's rest frame observer moves with velocity(v,0,0) light from the source travels with velocityu=(ux,uy,uz) 𝜔 is the angle between u and x axis tan𝜔=u2y+u2z

ux sin𝜔=u2y+u2z

c cos𝜔=ux

c 𝜃 is the angle between -u and x' axis𝜃+𝜔=𝜋𝜃=𝜋-𝜔 -in observer's rest frame source moves with velocity(-v,0,0) light from the source travels with velocityu'=(u'x,u'y,u'z) 𝜏 is the angle between u' and x axistan𝜏=u'2y+u'2z

u'x u'=𝛤((-v+ux)x'+uyz

𝛾)(u'x,u'y,u'z)=𝛤(-v+ux,uy/𝛾,uz/𝛾) tan𝜏=(𝛤uy

𝛾)2+(𝛤uz

𝛾)2

𝛤(-v+ux) tan𝜏=(𝛤

𝛾)2(u2y+u2z)

𝛤(-v+ux) tan𝜏=𝛤

𝛾u2y+u2z

𝛤(-v+ux) tan𝜏=u2y+u2z

𝛾(-v+ux) tan𝜏=u2y+u2z

c

𝛾(-v

c+ux

c) tan𝜏=sin𝜔

𝛾(-𝛽+cos𝜔) 𝜙 is the angle between -u' and x' axis𝜙+𝜏=𝜋𝜙=𝜋-𝜏 -tan𝜙=sin𝜃

𝛾(-𝛽-cos𝜃) tan𝜙=sin𝜃

𝛾(𝛽+cos𝜃) <Terrell-Penrose Effect>P(t,x,y,z)P'(t',x',y',z') O(0,0,0,0)O'(0,0,0,0) ct'=𝛾(ct-𝛽x)x'=𝛾(x-𝛽ct)y'=yz'=z P'(𝛾(t-vx

c2),𝛾(x-vt),y,z) for light generated at P to reach O=for light generated at P' to reach O' c(0-t)=-ct=x2+y2+z2≡rt=- r

ct'=- r'

c P'(𝛾(- r

c-vx

c2),𝛾(x+vr

c),y,z)P'(-𝛾

c(r+𝛽x),𝛾(x+𝛽r),y,z)t'=- r'

c=-𝛾

c(r+𝛽x) r'=𝛾(r+𝛽x)x'=𝛾(x+𝛽r) 𝛽r'=𝛾(𝛽r+𝛽2x)𝛽x'=𝛾(𝛽x+𝛽2r) 𝛾(1-𝛽2)r=r'-𝛽x'r=𝛾(r'-𝛽x') 𝛾(1-𝛽2)x=x'-𝛽r'x=𝛾(x'-𝛽r') x=𝛾(x'-𝛽𝛾(r+𝛽x))x=𝛾x'-𝛽𝛾2r-𝛽2𝛾2x x

𝛽𝛾=x'

𝛽-𝛾r-𝛽𝛾x(1

𝛽𝛾+𝛽𝛾)x=x'

𝛽-𝛾r1

𝛽𝛾+𝛽𝛾≡Ax'

𝛽≡BAx-B=-𝛾x2+y2+z2(Ax-B)2=𝛾2(x2+y2+z2)A2x2-2ABx+B2=𝛾2x2+𝛾y2+𝛾z2(A2-𝛾2)x2-2ABx+B2-𝛾y2-𝛾z2=0A2-𝛾2≡CCx2-2ABx+B2-𝛾y2-𝛾z2=0C(x2-2AB

Cx+B2

C)-𝛾y2-𝛾z2=0C(x2-2AB

Cx+A2B2

C2-A2B2

C2+B2

C)-𝛾y2-𝛾z2=0C(x-AB

C)2-𝛾y2-𝛾z2=A2B2

C-B2C(x-AB

C)2-𝛾y2-𝛾z2=B2(A2-C)

C C2

B2(A2-C)(x-AB

C)2-C𝛾

B2(A2-C)y2-C𝛾

B2(A2-C)z2=1 A2-C=𝛾2C𝛾=𝛾(A2-𝛾2)=𝛾(𝛽2𝛾2+1

𝛽2𝛾2+2-𝛾2)=𝛾((𝛽2-1)𝛾2+2+1

𝛽2𝛾2)=𝛾(-(1-𝛽2)(1-𝛽2)-1+2+1

𝛽2𝛾2)=𝛾(1+1

𝛽2𝛾2)=1

𝛽2𝛾+𝛾=1

𝛽(1

𝛽𝛾+𝛽𝛾)=A/𝛽 C2

B2𝛾2(x-AB

C)2-A/𝛽

B2𝛾2y2-A/𝛽

B2𝛾2z2=1 therefore, a plane of same x' valuelooks like a hyperboloid in reference frame S Hyperboloid z=-rz=-(𝛽z+w)2+d2z2=(𝛽z+w)2+d2z2=𝛽2z2+2w𝛽z+w2+d2(𝛽2-1)z2+2w𝛽z+w2+d2=0z=-w𝛽+w2𝛽2-(𝛽2-1)(w2+d2)

𝛽2-1z=w𝛽-w2𝛽2+(1-𝛽2)(w2+d2)

1-𝛽2 3d minkowski diagram