Generalizing the Klein-Gordon equation

Generalizing the Klein-Gordon equationKevin GibsonAugust 16, 2010IntroductionThe Klein-Gordon equation can be expressed as follows 1∇ 2 − 1 ∂ 2 c 2 ∂ t =− E 0 (1)2The purpose of this paper is twofold. First, show that the classical Schrödinger equationobtains from the Klein-Gordon equation in the classical limit of the kinetic energy beingmuch less than the rest energy of a particle. Having done this focus will then be turnedto generalizing the Klein-Gordon equation. This will be done for the case of a fourdimensionalspace then generalized to include higher dimensions.c ħ2Momentum, Operators & relativityIntroduce a scale factor h αβ for each coordinate defined according to its relation to themetric, h 2 αβ ≡ g αβ , with α and β indices that span all the coordinates. Furthermore theabbreviation h α ≡ h αα will be used. In Cartesian coordinates x α and h α are as follows 2 :x ={x , y , z ,t }(2a)h ={1, 1, 1,ic}(2b)In special relativity the momentum four-vector 3 can be expressed asp={ p , p , p , i E Dx y zc } (3)E D is defined as the dynamical energy, or the sum of the rest and kinetic energy. Theimaginary number i is used so the usual dot product of p with itself takes a similar formto the mass-shell relation.In fact one can also define p in the relativistic limit.p =m⋅L ⋅ ∂ x (4)∂tWith m is the relativistic mass and L α the scale factors for a space free of any forces. Forthe case of special relativity L t = ic. In this case we get for the time component ofmomentum1.

p t=m⋅i c= E Dc ⋅i c=i⋅E D2 cConnecting the Klein-Gordon equation to the Schrödinger EquationThis section will prove that the Klein-Gordon equation above becomes the Schrödingerequation for a free particle under the condition that the dynamical energy isapproximately equal to the rest energy. For now assume a four-dimensional Minkowskispace.The Klein-Gordon equation can be reworked as follows− ħ2⋅∇2m 2 ħ 22 i ħ⋅∂ =i ħ⋅∂∂ t ∂ t2E 0 ⋅∂2 ∂ t 2 E 0With E 0 = m rest·c 2 . It can be shown above that a fixed energy is possible under thecondition.i ħ⋅ ∂ ∂t =E D (6)In the classical limit E D ≈ E 0 and soi ħ⋅ ∂ ∂t ≈ E 0∂∂ t ≈ E 0i ħ (7)Under this approximation the Klein Gordon equation simplifies to(8)− ħ22 m ⋅∇ 2 E 0=i ħ⋅ ∂∂tThis is the Schrödinger equation with the rest energy included. By comparison with theclassical conservation of energy the total energy (kinetic plus rest plus potential energy)E is represented by the portion on the right side. It is therefore postulated that the totalenergy operator E will always be simplyE op =i ħ⋅ ∂ ∂t (9)so the relativistic quantum mechanical equations go to the Schroedinger equation in theclassical limit.Generalizing the Klein-Gordon EquationThe Laplacian for the space described by the scale factors in (2b) is 4□ 0 2 = ∂2 ∂ x 2 ∂2 ∂ y 2 ∂2 ∂ z 2 − 1 c 2⋅∂2 ∂ t 2 (10)(5)2.

Again, the zero subscript denotes a special relativistic coordinate system where noforces are present.In this light the Klein-Gordon equation above can be rewritten as□ 2 0=− E 20cħ (11)Which suggests a generalization of the Klein-Gordon equation as follows□ =− 2 E 20cħ (12)However as we allow for higher dimensions this will not reduce down to the originalKlein-Gordon equation for the case of a particle in a free space. To correct for this thegeneralized Klein-Gordon equation can be modified as follows□ [ 2 ∇ 2 0− 1 ∂ 2 c 2 ∂ t ] =− E 202 cħ □ 2 0 (13)With the del operator involving only the three spatial coordinates and the L denoting thatthe geometry of a free space is to be used.Black HolesThe above will lastly be applied to the case of black holes. Here the geometry of spacetimeis given by the Schwarzchild metric 5 .ds 2 =dr21− R sR1− sr r⋅d 2 r sin ⋅d 2 −c⋅t ⋅ 2r (14)R s is the Schwarzchild radius. The non-zero scale factors obtained from theSchwarzchild metric along with the product of all factors are as followsh r1R1−srh θrh φr⋅sin h tΠi c⋅ 1− R sri c r 2 ⋅sin 3.

Using (13) we can obtain the Klein-Gordon equation for black holes.1r 2⋅ ∂∂ r { r 2 −r⋅R S } ∂∂ r 1 r ∂ ⋅ ∂ ∂sin ∂ { 1r sin } ∂2 ∂ r 2c 2 ⋅r−R S ⋅∂2 ∂t =− E 20 2 c ħ(15)One important thing to note is the time derivative term will blow up at R s unless the timederivative goes to zero. Thus a wave packet moving into a black hole must freeze up atthe Schwarzchild radius, consistent with general relativistic considerations for objectsfalling into a Black Hole.ConclusionsThe Klein-Gordon equation and uncertainty principle can easily be generalized toaccommodate whatever space a particle may be in. However, it must be noted that thispaper does not address how to include particle-particle interactions in a relativisticmanner.Works Cited1. http://en.wikipedia.org/wiki/Klein-Gordon_equation2. Kevin Gibson Space-time Math (unpublished)www.mc.maricopa.edu/~kevinlg/i256/Relativity_Math.pdf3. John J. Brehm and William J. Mullin Introduction to the structure of Matter p. 634. Herbert Goldstein Classical Mechanics 2 nd ed. p. 3035. http://en.wikipedia.org/wiki/Schwarzschild_metricContact informationkevin.gibson@asu.edukevinlg@mesacc.edu4.