Spacetime diagrams. Assumes c=1.

Version 0.4
To do:
(i) Better/faster wavy light signal.
(ii) Improve interval plotter to better handle purely timelike (vertical) intervals.
(iii) Construct Lorentzian inner product to simplify point definitions (so I can use vectors)
(iv) Express Lorentz transform in matrix form as well?
(v) Add ...


Velocity of S' relative to S.

V=slider([-1,1,40])

Rapidity

function(g,s)=1/sqrt(1-s^2)

Lorentz transformation – space, time, longitudinal velocity.

function(X,a,b,V)=function(g,V)*[a-(V*b)],function(T,a,b,V)=function(g,V)*[b-(V*a)],function(B,b,V)=(b-V)/(1-(b*V))

Invariant interval.

function(D,X,T)=T^2-X^2

function(S,X,T,Y,U)=[T-U]^2-[X-Y]^2

Worldline plotter – (a,b) = point through which wordline passes; B = velocity.

function(W,a,b,B,X)=(X-a)/(B+d)+b

d=sn(1,-6)

Interval plotter. Not currently good at vertical (b=B) intervals.

function(I,a,b,A,B,X)=branch(if((B-b)/(A-a+d)*[X-a]+b,a<X<A),if((B-b)/(A-a+d)*[X-a]+b,A<X<a))

Light signal plotter – a,b = initial x,t; l = length of signal. Negative l is left-moving.

function(L,a,b,l,X)=branch(if(X-a+b,a<X<a+l/sqrt(2)),if(-X+a+b,a+l/sqrt(2)<X<a),if(0,l=0))

Wavy light signal plotter [write 0=K]

function(K,a,b,l,X,Y)=branch(if(Y-X+0.1*cos([15*[X+Y]])-a+b,a<X<a+l/sqrt(2)),if(-Y-X+0.1*cos([15*[X-Y]])+a+b,a+l/sqrt(2)<X<a))

S's coordinate axes in S'

function(W,0,0,1/V,prime(x))

function(W,0,0,V,prime(x))

function(W,function(X,0,b,V),function(T,0,b,V),1/V,prime(x)),b=set(-5,-4,-3,-2,-1,1,2,3,4,5,0)

function(W,function(X,a,0,V),function(T,a,0,V),V,prime(x)),a=set(-5,-4,-3,-2,-1,1,2,3,4,5,0)

S''s coordinate axes in S

function(W,0,0,-1/V,x)

function(W,0,0,-V,x)

function(W,function(X,0,b,-V),function(T,0,b,-V),-1/V,x),b=set(-5,-4,-3,-2,-1,1,2,3,4,5,0)

function(W,function(X,a,0,-V),function(T,a,0,-V),-V,x),a=set(-5,-4,-3,-2,-1,1,2,3,4,5,0)

Invariant hyberbolae through origin.

0=function(D,x,y)

0=function(D,prime(x),prime(y))

Invariant hyberbolae. (Twitchy; must have H on RHS.)

H=slider([0,4])

0=function(D,x,y)-H^2

0=function(D,prime(x),prime(y))-H^2

Light signals. (Happen to be from event (a,b) specified in S at the moment.)

s=slider([-5,5])

function(L,a,b,s,x)

function(L,function(X,a,b,V),function(T,a,b,V),s,prime(x))



Event (a,b) in S and its image in S' [or vice-versa]; invariant hyberbola through it. Worldline through (a,b) with velocity V1. Events simultaneous with (a,b) in S/S'.

a=slider([-2,2,40])

b=slider([-2,2,40])

V_1=slider([-1,1,40])

(a,b) in S and its image in S'...

vector(x,y)=vector(a,b)

vector(prime(x),prime(y))=vector(function(X,a,b,V),function(T,a,b,V))

0=function(D,x,y)-function(D,a,b),y>0

0=function(D,prime(x),prime(y))-function(D,function(X,a,b,V),function(T,a,b,V)),prime(y)>0

function(W,a,b,V_1,x)

function(W,function(X,a,b,V),function(T,a,b,V),function(B,V_1,V),prime(x))

function(W,function(X,a,b,V),function(T,a,b,V),1/V,prime(x))

(a,b) in S' and its image in S...

vector(prime(x),prime(y))=vector(a,b)

vector(x,y)=vector(function(X,a,b,-V),function(T,a,b,-V))

function(W,a,b,V_1,prime(x))

function(W,function(X,a,b,-V),function(T,a,b,-V),function(B,V_1,-V),x)

function(W,function(X,a,b,-V),function(T,a,b,-V),-1/V,x)

Event (p,q) in S and its image in S' [or vice-versa]. Worldline through (a,b) with velocity V2.

p=slider([-2,2,40])

q=slider([-2,2,40])

V_2=slider([-1,1,40])

(p,q) in S and its image in S' ...

vector(x,y)=vector(p,q)

vector(prime(x),prime(y))=vector(function(X,p,q,V),function(T,p,q,V))

function(W,p,q,V_2,x)

function(W,function(X,p,q,V),function(T,p,q,V),function(B,V_2,V),prime(x))

(p,q) in S' and its image in S ...

vector(prime(x),prime(y))=vector(p,q)

vector(x,y)=vector(function(X,p,q,-V),function(T,p,q,-V))

function(W,p,1,V_2,prime(x))

function(W,function(X,p,q,-V),function(T,p,q,-V),function(B,V_2,-V),x)

Interval connecting event (a,b) to (p,q) specified in S and its image in S'....

function(I,a,b,p,q,x)

function(I,function(X,a,b,V),function(T,a,b,V),function(X,p,q,V),function(T,p,q,V),prime(x))

0=function(D,prime(x),prime(y))-function(D,function(X,a,b,V),function(T,a,b,V))

0=function(D,prime(x),prime(y))-function(D,function(X,p,q,V),function(T,p,q,V))

Interval connecting event (a,b) to (p,q) specified in S' and its image in S....

function(I,a,b,p,q,prime(x))

function(I,function(X,a,b,-V),function(T,a,b,-V),function(X,p,q,-V),function(T,p,q,-V),x)


Length contraction. Rod at rest in S'. Worldines of ends of rod. Length of rod in S is separation between worldlines (ends of rod) at same moment of time in S. Length is greatest in rest frame of rod (S').

function(I,0,0.05,1,0.05,prime(x))

function(I,function(X,0,0.05,-V),function(T,0,0.05,-V),function(X,1,0.05,-V),function(T,1,0.05,-V),x)

function(W,function(X,0,0.05,-V),function(T,0,0.05,-V),V,x)

function(W,function(X,1,0.05,-V),function(T,1,0.05,-V),V,x)


Time dilation. Clock at rest in S'. Time between ticks of clock is shortest in rest frame of clock (S').

function(I,0.05,0,0.07000000000000001,1,prime(x))

function(I,function(X,0.05,0,-V),function(T,0.05,0,-V),function(X,0.07000000000000001,1,-V),function(T,0.07000000000000001,1,-V),x)





Author: David A. Craig <http://web.lemoyne.edu/~craigda/>

Graph of the formula

This file was created by Graphing Calculator 3.5.
Visit Pacific Tech to download the helper application to view and edit these equations live.