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Spacetime vector analysis
Volume 84A, number 2
PHYSICS LETTERS
13 July 1981
SPACETIME VECTOR ANALYSIS G. SOBCZYK Institute for Theoretical Physics, University of Wroctaw, Wroctaw, Poland Received 22 January 1981 Revised manuscript received 7 April 1981
Ordinary Gibbs—Heaviside vector algebra is complexified to apply to spacetime. The resulting algebra is isomorphic to both the Pauli algebra, and to the algebra of complex quatemions. Each inertial system is distinguished by a rest frame of real vectors. The rudiments of a spacetime vector analysis are given.
I a1
The complex vector algebra used in this article has been developed from a set of axioms in refs. [1] and
xB)oc~iH3’
[2].LetQ~~beacomplexthree-dimensionalvector space. The complexvectorsA,B, CEQ2c have the form 3
(A
A=akEkiE~ cskEk, B”f3~’Ek, c=ykEk, k=1 where ak, ~ 3k,7k are complex scalars of the forms
and
+itECforrealnumberss,t.Wegive~V~ thestructure of a complex three-dimensional euclidean space by defining the complex scalar product
71
a2
a3
~2 ~2
~3 73
(3) Ao(BX C)EC,
A X (B X C) =(A oB)C — (A 0 C)B 9c =(A XB)X C-fB X(A X C)E~
(4)
Except for a factor of i in (2) and (3), and consequently a difference of sign in (4), the complex vector
A 0 B = ~k13k = ak~3k= B o A E C~
where the summation over k = 1,2,3 is assumed. With respect to the complex scalar product the complex vectors {Ek) form an orthonormal basis of Q-’~,and upper and lower indices can be interchanged as a matter of convenience, Next, we define the complex vector product E1 A XB=i
a’ ~1
E2
identities (1 )—(4) have exactly the same form as the corresponding identities from the real Gibbs—Heaviside vector algebra. There is an even more fundamental product that can be constructed from (1) and (2) called the geometric product. The geometric product of A, B E defined by ABAoB+AXB, (5)
E31
2
a3
a~2
~3
=
B XA E
,
(2)
which turns tVC into a complex vector algebra. There are two complex triple products in ~ which can be constructed from (1) and (2):
is formal sum of the complex scalar E the C, and the complex vector part A X B part E~ A oB and has no direct analogy in the real Gibbs—Heaviside vector algebra. The geometric product is more fundamental than either (1) or (2), because both A oR andA X B can be defined in terms of the geometric product; that is (6) AoB~(AB+BA) and AXB~(AB-BA).
0 031—9163/81/0000—0000/S 02.50 © North-Holland Publishing Company
45
Volume 84A, number 2
PHYSICS LETTERS
In addition, the geometric product satisfies the important associative law of multiplication
Suppose that L: ~ satisfies
(7)
(AB)C=A(BC).
L(A)oL(B)AoB
Now let d = ~Ek} be an arbitrary oriented orthonormal frame of complex vectors, by which we mean EJOEk ~jk
forj,k
1,2.3,
=
E
1E2E3
=
i.
—
,
-~
~Vc is complex linear and
forallA,BEQ2c.
Then L(A) is said to be a Lorentz rotation. In ref. [1] it is shown that there exists a complex vector CE~l9c such that L(A)=e~’Ae~’, forallAEc)9
By the inertial system d , we mean d together with the operation of conjugation in ~ defined by ,
A =cskEk
13 July 1981
for allA =a Ek E~)~
C If C = ~ iêO ,where ë is a unit space vector of the inertial system d , then L(A) is a rotation through the
(8)
where &k is the usual complex conjugate of the cornplex scalar ak E C. A complex vector B E c~9Cis said to be a (real) space vector of the inertial system e iff
angle 0 of the component of A in the plane normal to the space vector ~. A Lorentz rotation is thus seen to be a generalization of the notion of a rotation in a plane. It might also be helpful to note the Euler identi-
B=B~~ ~k,
ty eiC0= cos(O)
fork
1,2,3.
(9)
—
—
To an observer at rest in d
the space vectors of ~ have the usual geometric interpretation line segments. IfB is a real space vector of of directed d ,then C E iB is said to be the imaginary space vector or space bivector of d dual to B. The space bivectors of d satisfy the property C=iB=—iB=—iB=—C (10)
+ iê
sin(O)
2
,
in the complex number plane of the bivector = i2c2 = —1].
ié[(ie)
If ~ and C are two inertial systems, then there exists a complex vector V such that V = V = V, and B = eVBe_ ~‘ , for all B ~ (11) .
The complex vector V is directly related to the velocity IS of the second inertial system ~‘ as seen in ~ ; this relationship will shortly be given. Writing V = I V I V, the Euler identity takes the form v = cosh Vt + V sinh 1 V I
and have the geometric interpretation of directed plane segments. The orthonormal frame {Ek} of real space vectors of d satisfy the multiplication rules E 1E2
=
E1 X E2
—
=
iE3
—
E3E 1 —L 3 XE 1 -iE 2
E2E3
,
E
E2 X E3 —
1E2E3—
F
~°(
=
iF1
The Lorentz rotation defined by (ll)is said to be
,
E 2XL3)—i, —
as can be easily verified by employing (2), (3) and (5). The geometric interpretation of these quantities is given in figure 1.
I
.
N
,‘
A
E
-
-
=
1 23
(12)
is a real orthonormal frame of s~’called the boost of {Ek}. See ref. [2] for proofs of(l 1) and (12), and for
and~xECJ9c}.
,‘~
(13)
A point X = ct +x E 9(will be called an event; twill
- -
Fig. 1. Geometric interpretation of the inertial system
46
then {E~}defined by = e V/2Eke_- V/2 for k
transformations in complex vector algebra. — By the spacetime horizon of the inertial system d we mean the set ~ {Xct+x ,tis a real scalar
.
E~E~E3 ~
the proper Lorentz rotation relating the inertial systems, —, . — ~ and ~ . If ~Ek } is a real orthonormal frame of ~,
a complete discussion of conjugations and Lorentz
,
I
.
.
be said to be the time of the event, andx itsplace. By the
Volume 84A, number 2
PHYSICS LETTERS
origin of the inertial system Cwe mean the event X = 0. By the space reversion of the event X = Cr + x, we mean the event X ct — x. In terms of the oper-
ation of space reversal, we define the spacetime interval (14) XX = (ct +x)(ct x) = c2t2 —x2 between the event X and the origin 0. Now suppose that~’is the spacetime horizon of a second inertial system By the universal mapping U: ~ we mean the mapping X’ = U(X) Xe— V for all XE ~(, (15) —
~‘.
-9-~’,
,
where Vis specified in (11). The reversal (X’) of X’ is related to the reversal X of X by
u/c
=
13 July 1981
tanh(V)
which directly relates the complex vector V defining the proper Lorentz rotation (11)to the relative velocity u of the observer as seen in d To complete these ideas we define thegradient operator of the inertial system C to be the usual gradient of vector analysis: ~7ss Ekak ,where a,~, aXk and x x’~’Ekis the position vector in C. Let ao = c1a/at be the time derivative of C. Let ~ = a/at be the time derivative of C. Using the gradient and time derivative, Maxwell’s equation in C takes the form (~ 0+ V)(E + ill) = p J, (20) —
(X ‘)
=
(Xe— V)_
=
(e V)_X_
=
e vX—
where F =E + iB is the electromagnetic field as expressed in the inertial system C, and p J is its source. For more of this to expression of Maxwell’s equation, and itsdetails relationship more usual formulations, see ref. [3]. Maxwell’s equation can be expressed in the second inertial system by multiplying both sides of (20) on the left by eV to obtain —
from2which — x’2 it = trivially X’(X’) follows = XX that = (ct)2
—
.
(ct’)
Thus, the spacetime interval between two events is independent of the inertial system in which it is measured. We have made the assumption that ~ is an inertial system. This means that an observer at rest in C experiences no acceleration. LettingX(t) = ct +x(t) be the position of the observer in ~( as a function oft, the statement that the observer is “at rest” in ~ translates into the mathematical equation dX/dtc’n’dx/dt~D(t)0.
(16)
Suppose now that an observer is at rest in a second inertial system C’. By using (15) we can write ci” +x’(t’) = [Ct +x(t)]e_V , (17) which relates the equations of motion of the observer in the inertial systems Cand Differentiating both 1d/dt’ and using the fact that sides by c u’(t’) of(17) = 0, gives ~‘.
1
=
dt/dt’ + c~(dt/dt’)(dx/dt) (18)
t’
+ v’)(E’ + iB’) = —J’ where (~3~ + V’)~e v(a 0 + V), F ssE + iB ssE’ + iB’, and p’ —J’ e V(p —J). Whereas complex vector algebra directly expresses the relative geometry of each inertial system, it is possible to go over to a four-dimensional Minkowski space in which the basic equations of physics find an absolute ~‘
expression independent of inertial systems. Let {Ek} be an orthonormal basis of the splitting complexof vector algebra tVc. By the Dirac—Clifford spacetime along {Ek}, we mean the factorization of the basis Ek, i.e., Ek
=
ek A e
= ekeo , fork 1,2,3of, a Dirac— (21) to obtain an0 orthonormal basis= ~e Clifford algebra7 .The basis {e,j~satisfies the rela}
tionships e~=l=_e~=_e~=_e 32,
=j3(1 +u/c)e_V,
where dt/dt’ and u dx/dz’. By using the operation of reversion, we can solve (18) for 13, finding the wellknown expression 2].ss~j3r~[I —u2/c2]~/2 (19) 1 rrj32[1 _(u/c) It also follows from (18) that
~
forp’v,
13
.
and E1E2E3e0e1ë2e3e0
Ae1 ,s.e2 Ae3=i.
The last relationship shows that the imaginary unit i has the geometric interpretation of the unit pseudo47
Volume 84A, number 2
PHYSICS LETTERS
scalar of both the complex vector algebra ~ c and the Dirac—Clifford algebra ~ . The algebra ~7Dis seen to 9c as an even subalgebra known as encompass C ~~ the Pauli subalgebra [2,3]. The operation of reversal used in this article corresponds to the main antiautomorphism in the Dirac—Clifford algebra ~D.
On the level of the Dirac algebra, every event in absolute spacetime is labeled by a Dirac vector x =xMe~(summation p = 0,1,2,3). Inertial systems are then identified with constant time-like unit vector fields. If e 0 is the unit time-like vector (e~= 1) of the inertial system C ,as defined in (13), then the event in 9( corresponding to the event x in absolute spacetime is X ssxe0
=
x e0 +x Ac0
=
ci’ +x
.
(22)
Hestenes has extensively developed this approach in a series of papers, e.g., refs. [4,5]. Finally, we remark that the complex vector algebra used in this paper has already demonstrated its power
48
13 July 1981
by providing new insight into the algebraic structure of important tensors in physics [6,7]. In another article [8], we show how the algebraic classification of the electromagnetic, conformal Weyl, and energy—snomentum tensors reduces to the classification of a single general linear operator in CJ9 c~ References [1] G. Sobczyk, A complex Gibbs—Heaviside vector algebra for spacetime, Acta Phys. Pol. (April 1981), to be published. [2] G. Sobczyk, Conjugations and hermitian operators in Acta Phys. Pol. B12 (1981). 131 spacetime, D. Hestenes, Space—time algebra (Gordon and Breach, New York, 1966). [4] D. Hestenes, J. Math. Phys. 15 (1974) 1768. [51D. Hestenes, Am. J. Phys. 47 (1979) 5. [61 G. Sobczyk, Spacetime algebra approach to curvature, J. Math. Phys. (January 1981), to be published. [7] G.Sobczyk,ActaPhys.Pol.Bli(1980)no.8. [8] G. Sobczyk, following letter.
PHYSICS LETTERS
13 July 1981
SPACETIME VECTOR ANALYSIS G. SOBCZYK Institute for Theoretical Physics, University of Wroctaw, Wroctaw, Poland Received 22 January 1981 Revised manuscript received 7 April 1981
Ordinary Gibbs—Heaviside vector algebra is complexified to apply to spacetime. The resulting algebra is isomorphic to both the Pauli algebra, and to the algebra of complex quatemions. Each inertial system is distinguished by a rest frame of real vectors. The rudiments of a spacetime vector analysis are given.
I a1
The complex vector algebra used in this article has been developed from a set of axioms in refs. [1] and
xB)oc~iH3’
[2].LetQ~~beacomplexthree-dimensionalvector space. The complexvectorsA,B, CEQ2c have the form 3
(A
A=akEkiE~ cskEk, B”f3~’Ek, c=ykEk, k=1 where ak, ~ 3k,7k are complex scalars of the forms
and
+itECforrealnumberss,t.Wegive~V~ thestructure of a complex three-dimensional euclidean space by defining the complex scalar product
71
a2
a3
~2 ~2
~3 73
(3) Ao(BX C)EC,
A X (B X C) =(A oB)C — (A 0 C)B 9c =(A XB)X C-fB X(A X C)E~
(4)
Except for a factor of i in (2) and (3), and consequently a difference of sign in (4), the complex vector
A 0 B = ~k13k = ak~3k= B o A E C~
where the summation over k = 1,2,3 is assumed. With respect to the complex scalar product the complex vectors {Ek) form an orthonormal basis of Q-’~,and upper and lower indices can be interchanged as a matter of convenience, Next, we define the complex vector product E1 A XB=i
a’ ~1
E2
identities (1 )—(4) have exactly the same form as the corresponding identities from the real Gibbs—Heaviside vector algebra. There is an even more fundamental product that can be constructed from (1) and (2) called the geometric product. The geometric product of A, B E defined by ABAoB+AXB, (5)
E31
2
a3
a~2
~3
=
B XA E
,
(2)
which turns tVC into a complex vector algebra. There are two complex triple products in ~ which can be constructed from (1) and (2):
is formal sum of the complex scalar E the C, and the complex vector part A X B part E~ A oB and has no direct analogy in the real Gibbs—Heaviside vector algebra. The geometric product is more fundamental than either (1) or (2), because both A oR andA X B can be defined in terms of the geometric product; that is (6) AoB~(AB+BA) and AXB~(AB-BA).
0 031—9163/81/0000—0000/S 02.50 © North-Holland Publishing Company
45
Volume 84A, number 2
PHYSICS LETTERS
In addition, the geometric product satisfies the important associative law of multiplication
Suppose that L: ~ satisfies
(7)
(AB)C=A(BC).
L(A)oL(B)AoB
Now let d = ~Ek} be an arbitrary oriented orthonormal frame of complex vectors, by which we mean EJOEk ~jk
forj,k
1,2.3,
=
E
1E2E3
=
i.
—
,
-~
~Vc is complex linear and
forallA,BEQ2c.
Then L(A) is said to be a Lorentz rotation. In ref. [1] it is shown that there exists a complex vector CE~l9c such that L(A)=e~’Ae~’, forallAEc)9
By the inertial system d , we mean d together with the operation of conjugation in ~ defined by ,
A =cskEk
13 July 1981
for allA =a Ek E~)~
C If C = ~ iêO ,where ë is a unit space vector of the inertial system d , then L(A) is a rotation through the
(8)
where &k is the usual complex conjugate of the cornplex scalar ak E C. A complex vector B E c~9Cis said to be a (real) space vector of the inertial system e iff
angle 0 of the component of A in the plane normal to the space vector ~. A Lorentz rotation is thus seen to be a generalization of the notion of a rotation in a plane. It might also be helpful to note the Euler identi-
B=B~~ ~k,
ty eiC0= cos(O)
fork
1,2,3.
(9)
—
—
To an observer at rest in d
the space vectors of ~ have the usual geometric interpretation line segments. IfB is a real space vector of of directed d ,then C E iB is said to be the imaginary space vector or space bivector of d dual to B. The space bivectors of d satisfy the property C=iB=—iB=—iB=—C (10)
+ iê
sin(O)
2
,
in the complex number plane of the bivector = i2c2 = —1].
ié[(ie)
If ~ and C are two inertial systems, then there exists a complex vector V such that V = V = V, and B = eVBe_ ~‘ , for all B ~ (11) .
The complex vector V is directly related to the velocity IS of the second inertial system ~‘ as seen in ~ ; this relationship will shortly be given. Writing V = I V I V, the Euler identity takes the form v = cosh Vt + V sinh 1 V I
and have the geometric interpretation of directed plane segments. The orthonormal frame {Ek} of real space vectors of d satisfy the multiplication rules E 1E2
=
E1 X E2
—
=
iE3
—
E3E 1 —L 3 XE 1 -iE 2
E2E3
,
E
E2 X E3 —
1E2E3—
F
~°(
=
iF1
The Lorentz rotation defined by (ll)is said to be
,
E 2XL3)—i, —
as can be easily verified by employing (2), (3) and (5). The geometric interpretation of these quantities is given in figure 1.
I
.
N
,‘
A
E
-
-
=
1 23
(12)
is a real orthonormal frame of s~’called the boost of {Ek}. See ref. [2] for proofs of(l 1) and (12), and for
and~xECJ9c}.
,‘~
(13)
A point X = ct +x E 9(will be called an event; twill
- -
Fig. 1. Geometric interpretation of the inertial system
46
then {E~}defined by = e V/2Eke_- V/2 for k
transformations in complex vector algebra. — By the spacetime horizon of the inertial system d we mean the set ~ {Xct+x ,tis a real scalar
.
E~E~E3 ~
the proper Lorentz rotation relating the inertial systems, —, . — ~ and ~ . If ~Ek } is a real orthonormal frame of ~,
a complete discussion of conjugations and Lorentz
,
I
.
.
be said to be the time of the event, andx itsplace. By the
Volume 84A, number 2
PHYSICS LETTERS
origin of the inertial system Cwe mean the event X = 0. By the space reversion of the event X = Cr + x, we mean the event X ct — x. In terms of the oper-
ation of space reversal, we define the spacetime interval (14) XX = (ct +x)(ct x) = c2t2 —x2 between the event X and the origin 0. Now suppose that~’is the spacetime horizon of a second inertial system By the universal mapping U: ~ we mean the mapping X’ = U(X) Xe— V for all XE ~(, (15) —
~‘.
-9-~’,
,
where Vis specified in (11). The reversal (X’) of X’ is related to the reversal X of X by
u/c
=
13 July 1981
tanh(V)
which directly relates the complex vector V defining the proper Lorentz rotation (11)to the relative velocity u of the observer as seen in d To complete these ideas we define thegradient operator of the inertial system C to be the usual gradient of vector analysis: ~7ss Ekak ,where a,~, aXk and x x’~’Ekis the position vector in C. Let ao = c1a/at be the time derivative of C. Let ~ = a/at be the time derivative of C. Using the gradient and time derivative, Maxwell’s equation in C takes the form (~ 0+ V)(E + ill) = p J, (20) —
(X ‘)
=
(Xe— V)_
=
(e V)_X_
=
e vX—
where F =E + iB is the electromagnetic field as expressed in the inertial system C, and p J is its source. For more of this to expression of Maxwell’s equation, and itsdetails relationship more usual formulations, see ref. [3]. Maxwell’s equation can be expressed in the second inertial system by multiplying both sides of (20) on the left by eV to obtain —
from2which — x’2 it = trivially X’(X’) follows = XX that = (ct)2
—
.
(ct’)
Thus, the spacetime interval between two events is independent of the inertial system in which it is measured. We have made the assumption that ~ is an inertial system. This means that an observer at rest in C experiences no acceleration. LettingX(t) = ct +x(t) be the position of the observer in ~( as a function oft, the statement that the observer is “at rest” in ~ translates into the mathematical equation dX/dtc’n’dx/dt~D(t)0.
(16)
Suppose now that an observer is at rest in a second inertial system C’. By using (15) we can write ci” +x’(t’) = [Ct +x(t)]e_V , (17) which relates the equations of motion of the observer in the inertial systems Cand Differentiating both 1d/dt’ and using the fact that sides by c u’(t’) of(17) = 0, gives ~‘.
1
=
dt/dt’ + c~(dt/dt’)(dx/dt) (18)
t’
+ v’)(E’ + iB’) = —J’ where (~3~ + V’)~e v(a 0 + V), F ssE + iB ssE’ + iB’, and p’ —J’ e V(p —J). Whereas complex vector algebra directly expresses the relative geometry of each inertial system, it is possible to go over to a four-dimensional Minkowski space in which the basic equations of physics find an absolute ~‘
expression independent of inertial systems. Let {Ek} be an orthonormal basis of the splitting complexof vector algebra tVc. By the Dirac—Clifford spacetime along {Ek}, we mean the factorization of the basis Ek, i.e., Ek
=
ek A e
= ekeo , fork 1,2,3of, a Dirac— (21) to obtain an0 orthonormal basis= ~e Clifford algebra7 .The basis {e,j~satisfies the rela}
tionships e~=l=_e~=_e~=_e 32,
=j3(1 +u/c)e_V,
where dt/dt’ and u dx/dz’. By using the operation of reversion, we can solve (18) for 13, finding the wellknown expression 2].ss~j3r~[I —u2/c2]~/2 (19) 1 rrj32[1 _(u/c) It also follows from (18) that
~
forp’v,
13
.
and E1E2E3e0e1ë2e3e0
Ae1 ,s.e2 Ae3=i.
The last relationship shows that the imaginary unit i has the geometric interpretation of the unit pseudo47
Volume 84A, number 2
PHYSICS LETTERS
scalar of both the complex vector algebra ~ c and the Dirac—Clifford algebra ~ . The algebra ~7Dis seen to 9c as an even subalgebra known as encompass C ~~ the Pauli subalgebra [2,3]. The operation of reversal used in this article corresponds to the main antiautomorphism in the Dirac—Clifford algebra ~D.
On the level of the Dirac algebra, every event in absolute spacetime is labeled by a Dirac vector x =xMe~(summation p = 0,1,2,3). Inertial systems are then identified with constant time-like unit vector fields. If e 0 is the unit time-like vector (e~= 1) of the inertial system C ,as defined in (13), then the event in 9( corresponding to the event x in absolute spacetime is X ssxe0
=
x e0 +x Ac0
=
ci’ +x
.
(22)
Hestenes has extensively developed this approach in a series of papers, e.g., refs. [4,5]. Finally, we remark that the complex vector algebra used in this paper has already demonstrated its power
48
13 July 1981
by providing new insight into the algebraic structure of important tensors in physics [6,7]. In another article [8], we show how the algebraic classification of the electromagnetic, conformal Weyl, and energy—snomentum tensors reduces to the classification of a single general linear operator in CJ9 c~ References [1] G. Sobczyk, A complex Gibbs—Heaviside vector algebra for spacetime, Acta Phys. Pol. (April 1981), to be published. [2] G. Sobczyk, Conjugations and hermitian operators in Acta Phys. Pol. B12 (1981). 131 spacetime, D. Hestenes, Space—time algebra (Gordon and Breach, New York, 1966). [4] D. Hestenes, J. Math. Phys. 15 (1974) 1768. [51D. Hestenes, Am. J. Phys. 47 (1979) 5. [61 G. Sobczyk, Spacetime algebra approach to curvature, J. Math. Phys. (January 1981), to be published. [7] G.Sobczyk,ActaPhys.Pol.Bli(1980)no.8. [8] G. Sobczyk, following letter.