1
("Deriving the Kaluza-Klein equation of motion.")$
2
("For reference, see http://www.vttoth.com/KK/kk.htm")$
4
("KALUZA.MAC contains definitions and contraction properties for G4 and G5")$
6
("Predeclaring some 4d indices")$
7
ASSUME(K<=4,L<=4,M<=4)$
9
("Equation of motion in empty 5-space")$
10
SHOW('DIFF(x([],[A]),t,2)+'CHR2([B,C],[A])*'DIFF(x([],[B]),t)*'DIFF(x([],[C]),t)=0)$
11
SHOW(PART(FIRST(%),1))$
12
SHOW(SUBST(M,C,%)+SUBST(5,C,%))$
13
SHOW(SUBST(L,B,%)+SUBST(5,B,%)+PART(FIRST(%TH(3)),2)=LAST(%TH(3)))$
15
("We are only interested in the case where A is a 4D index")$
16
SHOW(SUBST(K,A,%TH(2)))$
18
("We protect one of the Christoffel-symbols from expansion")$
19
SHOW(SUBST(CHR2KLM,'CHR2([L,M],[K]),%TH(2)))$
23
("Now we break this up into two parts depending on whether %1=5")$
24
MAP(LAMBDA([U],BLOCK(IF FREEOF(%1,U) THEN U ELSE U+SUBST(5,%1,U))),FIRST(%TH(2)))=LAST(%TH(2))$
30
("Now we're ready to isolate the electromagnetic field tensor")$
31
MAP(LAMBDA([U],FACTOROUT(U,G55)),%TH(2))$
32
SHOW(APPLY1(%,EVPOT))$
34
("Contracting and rearranging yields the equation in the usual form")$
40
SHOW(SUBST('CHR2([L,M],[K]),CHR2KLM,%))$
1
/* Copyright (C) 2004 Viktor T. Toth <http://www.vttoth.com/>
3
* This program is free software; you can redistribute it and/or
4
* modify it under the terms of the GNU General Public License as
5
* published by the Free Software Foundation; either version 2 of
6
* the License, or (at your option) any later version.
8
* This program is distributed in the hope that it will be
9
* useful, but WITHOUT ANY WARRANTY; without even the implied
10
* warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
11
* PURPOSE. See the GNU General Public License for more details.
13
* The equation of motion of a free particle in a five dimensional
14
* Kaluza-Klein metric appears as the motion of a charged particle
15
* in four dimensional space in the presence of an EM field
20
Deriving the Kaluza-Klein equation of motion.
21
For reference, see http://www.vttoth.com/KK/kk.htm")$
23
("We first load ITENSOR and set up the 5-dimensional metric.
24
We also set up contraction properties for both the 4-dimensional
25
and the 5-dimensional metric tensors.")$
27
if get('itensor,'version)=false then load(itensor);
28
(derivabbrev:true, dim:5, imetric:g5, defcon(g4),defcon(g5),
29
defcon(g4,g4,kdelta), defcon(g5,g5,kdelta))$
31
("To set up the metric components, we need some helper functions.
32
The function predval() determines if a predicate can be evaluated.
33
It returns false if the predicate would return an error. The
34
function difflist() applies the differential operator to elements
37
predval(prd):=block([retval,saved_prederror:prederror],
39
retval:ev(prd,pred)=true or ev(prd,pred)=false,
40
prederror:saved_prederror,
43
difflist(exp,lst):=if length(lst)=0 then exp
44
else difflist(idiff(exp,lst[1]),rest(lst))$
46
("Metric components are defined conditionally, allowing us to treat
47
the fifth index in a unique way.")$
49
a(l1,l2,[l3]):=if member(5,l3) then 0 else funmake('a,append([l1,l2],l3))$
50
g4(l1,l2,[l3]):=if member(5,l3) then 0 else funmake('g4,append([l1,l2],l3))$
52
if member(5,l3) then 0
55
if not (predval(l1[1]<=4) and predval(l1[2]<=4)) then
56
funmake('g5,append([l1,l2],l3))
57
else if l1[1]<=4 and l1[2]<=4 then
58
apply('g4,append([l1,l2],l3))+
59
g55*difflist(a([l1[1]],[])*a([l1[2]],[]),l3)
60
else if l1[1]<=4 then g55*apply('a,append([[l1[1]],[]],l3))
61
else if l1[2]<=4 then g55*apply('a,append([[l1[2]],[]],l3))
62
else if l3#[] then 0 else g55
66
if not (predval(l2[1]<=4) and predval(l2[2]<=4)) then
67
funmake('g5,append([l1,l2],l3))
68
else if l2[1]<=4 and l2[2]<=4 then apply('g4,append([l1,l2],l3))
69
else if l2[1]<=4 then -apply('a,append([[],[l2[1]]],l3))
70
else if l2[2]<=4 then -apply('a,append([[],[l2[2]]],l3))
71
else if l3#[] then sum(difflist(a([i],[])*a([],[i]),l3),i,1,4)
72
else 1/g55+sum(a([i],[])*a([],[i]),i,1,4)
74
else funmake('g5,append([l1,l2],l3))$
76
("Now we're ready to begin the analysis. First, we predeclare
77
some 4-dimensional indices:")$
78
assume(k<=4,l<=4,m<=4)$
80
("The equation of motion in empty 5-space:")$
82
ishow('diff(x([],[a]),t,2)+
83
'ichr2([b,c],[a])*'diff(x([],[b]),t)*'diff(x([],[c]),t)=0)$
84
ishow(part(first(%),1))$
85
ishow(subst(m,c,%)+subst(5,c,%))$
86
ishow(subst(l,b,%)+subst(5,b,%)+part(first(%th(3)),2)=last(%th(3)))$
88
("We are only interested in the case where A is a 4D index:")$
89
ishow(subst(k,a,%th(2)))$
91
("We protect one of the Christoffel-symbols from expansion:")$
92
ishow(subst(chr2klm,'ichr2([l,m],[k]),%th(2)))$
96
("Now we break this up into two parts depending on whether %1=5:")$
97
map(lambda([u],block(if freeof(%1,u) then u else u+subst(5,%1,u))),
98
first(%th(2)))=last(%th(2))$
104
("Now we're ready to isolate the electromagnetic field tensor:")$
105
map(lambda([u],factorout(u,g55)),%th(2))$
106
ishow(ratsubst(-f([%1,%2],[]),a([%1],[],%2)-a([%2],[],%1),%))$
108
("Contracting and rearranging yields the equation in the usual form:")$
113
EQ:subst('ichr2([l,m],[k]),chr2klm,%)$
42
116
("But what about the 5D Christoffel-symbol?")$
43
SHOW(CHR2([K,L],[M]))$
52
SCANMAP(LAMBDA([U],APPLY1( RATSIMP(U,G55,G4([],[%1,M]),A([K],[])) ,EVPOT)),%)$
57
RATSUBST(CHR42([K,L],[M]),G4([],[%1,M])*(G4([L,%1],[],K)+G4([K,%1],[],L)-G4([K,L],[],%1))/2,%)$
117
/*ishow(ichr2([k,l],[m]))$*/
118
ishow(ichr2([l,m],[k]))$
127
/*ratsubst(-f([%1,k],[]),a([%1],[],k)-a([k],[],%1),%)$*/
128
ratsubst(-f([%1,l],[]),a([%1],[],l)-a([l],[],%1),%)$
129
ratsubst(-f([%1,m],[]),a([%1],[],m)-a([m],[],%1),%)$
130
ishow(factor(contract(expand(%))))$
134
/*ratsubst(ichr42([k,l],[m]),
135
g4([],[m,%1])*(g4([l,%1],[],k)+g4([k,%1],[],l)-g4([k,l],[],%1))/2,%)$*/
136
ratsubst(ichr42([l,m],[k]),
137
g4([],[k,%1])*(g4([m,%1],[],l)+g4([l,%1],[],m)-g4([l,m],[],%1))/2,%)$
61
141
("The extra term is presumably the curvature caused by the EM field.")$
62
SHOW(MAP(FACTOR,COMBINE(DISTRIB(%TH(2)))))$
142
/*ishow('ichr2([k,l],[m])=map(factor,combine(distrib(%th(2)))))$*/
143
ishow('ichr2([l,m],[k])=map(factor,combine(distrib(%th(2)))))$
144
("Or, if you wish, you can apply this result to the equation of motion:")$
145
ishow(subst(rhs(%th(2)),lhs(%th(2)),EQ))$
147
/* End of demo -- comment line needed by MAXIMA to resume demo menu */
added
removed
share/tensor/kaluza.dem
