CITE THIS NOTEBOOK: Generalization of the Galilei- and Lorentz-transformations by Oliver Seipel. Wolfram Community JAN 3 2023.
If you want to know what a theorems means, you have to look what its proof proofs.
Ludwig Wittgenstein.
Ludwig Wittgenstein.
Motivation
Motivation
You may have seen the Galilei-Transformations:(G1) (G2) from classical physics.And the Lorentz-Transformations:(L1) (L2) from special theory of relativity with =.Together with the principle of relativity the addition theorem follows:(G3) (L3) And for the speed of light we have(G4) (L4)
x'=x-vt
x=x'+vt'
t'=t
t=t'
x'=γ(x-vt)
x=γ(x'+vt')
t'=γt-x
v
2
c
t=γt'+x'
v
2
c
1
γ
1-
2
v
2
c
u=u'+v
u=
u'+v
1+
u'v
2
c
c'=c-v
c’=c
The question of concern here is:
What are the general transformations derived from the principle of relativity alone (i.e. neither using like in special theory of relativity nor as in classical physics)?
What are the general transformations derived from the principle of relativity alone (i.e. neither using
c'=c
t'=t
This will be the answer:(S1) , (S2) and the addition theorem(S3) And for the speed of light(S4) for some constant . is (L4), is (G4)
x'=k(x-vt)
k=>0
1
1-1+
v
c
v
c'
t'=kt-k-
1
k
x
v
u=
u'+v
1+u'
k-
1
k
kv
c'=c-αv
0<=α<=1
α=0
α=1
k
In[]:=
Simplify/.cp->c
1
1-1+
v
c
v
cp
Out[]=
1
1-
2
v
2
c
k
In[]:=
Simplify/.cp->c-v
1
1-1+
v
c
v
cp
Out[]=
1
Remark:
The reference frame S has coordinates and reference frame S’ has coordinates , t’. Reference frame S’ moves with relativ to S and reference frame S moves with relativ to S’. Both systems are inertial frames of reference.
The reference frame S has coordinates
x,t
x'
v
-v
Derivation of the transformations (S1) and (S2)
Derivation of the transformations (S1) and (S2)
Remark: x’ is interpreted as Derivative:
In[]:=
FullForm[x']
Out[]//FullForm=
Derivative[1][x]
So we do some formatting trick:
In[]:=
Format[xp]:="x'"Format[tp]:="t'"Format[up]:="u'"Format[cp]:="c'"
In[]:=
principleofrelativity={v->-v,t->tp,tp->t,xp->x,x->xp,u->up,up->u,c->cp,cp->c}
Out[]=
{v-v,tt',t't,x'x,xx',uu',u'u,cc',c'c}
A straight uniform motion requires a linear transformation:
x'=x+t
k
1
k
2
The origin of S’ is moving with as seen from S.
x'=0
x=vt
In[]:=
Simplify[xp==x+t/.First[Solve[0==vt+t,]]]
k
1
k
2
k
1
k
2
k
2
Out[]=
x'(-tv+x)
k
1
So we really have only a single constant :
k
In[]:=
S1=xp==k(x-vt)
Out[]=
x'k(-tv+x)
Remark: You can change this equation into a rule by:
In[]:=
Rule@@S1
Out[]=
x'k(-tv+x)
With the principle of relativity it follows ( and must also hold, otherwise the principle of relativity is violated)
k=k'
In[]:=
S1p=S1/.principleofrelativity
Out[]=
xk(t'v+x')
For the time-transformation it follows:
In[]:=
S2p=Equal@@Apart[First[First[Solve[S1,t]]]/.Rule@@S1p]
Out[]=
tkt'+
(-1+)x'
2
k
kv
And with the principle of relativity:
In[]:=
S2=S2p/.principleofrelativity
Out[]=
t'kt-
(-1+)x
2
k
kv
Remark: otherwise the time could move backwards .
k>0
Relativity of simultaneity
Relativity of simultaneity
In[]:=
Format[Δxp]:="Δx'"Format[Δtp]:="Δt'"Format[Δx]:="Δx"Format[Δt]:="Δt"
In[]:=
replacewithdifferences={x->Δx,t->Δt,tp->Δtp,xp->Δxp}
Out[]=
{xΔx,tΔt,t'Δt',x'Δx'}
Time difference in S’ for vanishing time differences in S:
In[]:=
S2/.replacewithdifferences/.Δt->0
Out[]=
Δt'-
(-1+)Δx
2
k
kv
This is zero if =1. Only in classical physics all oberservers agree with simultaneity.
2
k
Relativity of length
Relativity of length
Space difference in S’ for vanishing time differences in S:
(We do measure the lenght at the same time in S’ only in classical physics.)
(We do measure the lenght at the same time in S’ only in classical physics.)
In[]:=
S1/.replacewithdifferences/.Δt->0
Out[]=
Δx'kΔx
Space difference in S for vanishing time differences in S’:
(We do measure the lenght at the same time in S only in classical physics.)
(We do measure the lenght at the same time in S only in classical physics.)
In[]:=
S1p/.replacewithdifferences/.Δtp->0
Out[]=
ΔxkΔx'
You have to be carfull at what conditions you are measuring the length!
Relativity of time
Relativity of time
Time difference in S’ for vanishing space differences in S: (i.e. a clock resting in S, moving in S’)
In[]:=
S2/.replacewithdifferences/.Δx->0
Addition of speed
Addition of speed
A body is moving in S on a straight line at constant speed:
The same body is moving in S’ on a straight line at constant speed:
We do some hand optimization:
They are really the same:
The speed of light in different reference frames
The speed of light in different reference frames
A lightning as seen from S:
And S’:
We solve the equation “Lightning” for t and insert it in S1:
We solve the equation “Lightningp” for t’ and insert it in S1’:
And optimized by a human:
But they are all the same:
To simplify we need the assumption that the speed of light is positive:
Now we parameterize the speed of light:
Classical physics:
Special Relativity:
Generalized physics:
some hand optimization:
They are really the same:
Addition of speeds for this case:
Ans from the general case above:
They are the same:
Fizeau 1851
Fizeau 1851
Fizeau measured the speed of light in moving water:
Comparing the coefficients gives:
But it’s not with classical physics:
The best value for α from Fizeau is: