It consists of conductivity of the medium, which is related to the friction of the system. Calculating black hole quasinormal modes by using poschl. On dark matter, spiral galaxies, and the axioms of general. The orthonormalization relation for the plane waves 10 is 11 12 14. We examine the einstein equation coupled to the klein gordon equation. It is secondorder in space and time and manifestly lorentzcovariant. The conductivity term can also feels the existence of the bulk matter through the particle density term n, where m ne w v 2. A dynamical equation that is invariant with respect to the transformations of coordinates is a poincar. The schrodinger equation applies to particles in motion at nonrelativistic speeds, while the klein gordon and dirac equations represent.
Relativistic quantum mechanics university of glasgow school of. Klein gordon equation alternatively, apply quantization to energymomentum invariant. In the general relativity theory, we find gravitational matter wave by klein gordon wave equation. Derived in the early part of the 20 th century, they describe the wave like motion of material subatomic particles. Hence, it predicts the existence of an intrinsic magnetic moment for the electron and gives its correct value. To compare the electromagnetic dirac equation with the electromagnetic klein gordon equation in a manner analogous to the scheme adopted in ref. A singularity theorem for einsteinkleingordon theory. The conservation of matter in general relativity project euclid. Pdf dirac and kleingordon equations in curved space. General relativity and quantum theory it is well recognized in physics that the two basic physics theories, general relativity and quantum theory, are each lacking in complete descriptions of reality. It is a quantized version of the relativistic energymomentum relation. The einsteinkleingordon equations, wave dark matter, and. Klein gordon equation tried an equation that was linear in the energy, mass and momentum.
Solving the maxwellkleingordon equation in the lattice. In this paper, we make free particles wave function as the solution of klein gordon equation in. Pdf modified general relativity and the kleingordon equation in. Generalized dirac and kleingordon equations for spinor. Fields institute centenary of einsteins equations of. Although weak, this is a significant difference that shows the influence of spin in.
In the lowfield, nonrelativistic limit, the einstein klein gordon system reduces to the poissonschrodinger system. Specially, this article is that quantization of gravitational wave is made by klein gordon wave. The klein gordon kg equation 4,5, one of the fundamental poincar. Is the klein gordon equation the same in all reference frames. We explore klein gordon equation in the new framework of quasihermitian quantum mechanics. Solutions to common problems with probability interpretation and inde. From einstein to kleingordon quantum mechanics and relativity. The plane waves 10 are basic solutions and the field 9 is constructed by a general superposition of the basic states. Specially, this article is that klein gordon wave equations is treated by. Derivation of kleingordon equation in general relativity. Derivation of maxwells equation for diffusion current and. Gordoneinstein kge equations, was first considered in the context. I could be wrong, but it seems to me that if you just take the flatspacetime version and replace partial derivatives with covariant derivatives, you ought to get something much prettier.
Modi ed general relativity and the kleingordon equation in. As we can see now, the apparent problems were manifestations of the new physics. Pdf klein gordon equation is a relativistic wave equation. In 1920s physicists were looking for relativistic quantum mechanic equation for the electron. Theorem dong, lefloch and wyatt 2018 consider the abelian u1 higgs mechanism coupled to a fermion with yukawa coupling. From the mass shell condition 6 this results in the kleingordon equation as the basic field equation of the scalar field.
The hydrodynamic representation of the kleingordon equation kge. Natural geometric axioms motivate studying the einstein klein gordon equations with a cosmological constant. Nov 16, 2017 in relativistic quantum mechanics this problem allows also to highlight the implications of special relativity for quantum physics, namely the effect that spin has on the quantized energy spectra. The klein gordon equation was historically rejected as a fundamental quantum equation because it predicted negative probability density. By using special relativity, one can derive a relativistic version of the schr. General relativity and geometry 230 9 lie derivative, symmetries and killing vectors 231 9. Covariant notation for the kleingordon equation if an equation is consistent with the principles of special relativity, it should be possible to write it in covariant form, that is, in a form that is the same in all lorentz frames. I go over fourvector notation, fourgradient, and writing the equ. The focus of this week will be on general nonlinear wave equations that are not directly related to the theory of relativity. The constant ground state is stable and the stability is uniform with respect to the fermion mass.
Pdf geometry and motion in general relativity james owen. While this geometric motivation is new, modeling dark matter. Pdf quantization of gravitational wave by kleingordon equation. The gravity of the classical kleingordon field mdpi. Equation 5 can further be divided into two types of equations, one considered as a constraint no time derivative, i. Each is correct and makes precise predictions when restricted to its own realm. The klein gordon equation is a secondorder differential equation in t and thus the initial values of. The klein gordon equation for a free eld with a particular spin in minkowski spacetime is. Therefore, as long as energy and momentum follows the einsteins relation eq. Kapitza spring 2018 second quantization of the kleingordon.
Kleingordon and dirac equations in general relativity. General relativity is the classical theory that describes the evolution of systems under the e ect of gravity. Neither explains interactions between gravitation and. Its history goes back to 1915 when einstein postulated that the laws of gravity can be expressed as a system of equations, the socalled einstein equations. Solution of the klein gordon equation the covariant generalization of the klein gordon equation in curved spacetime has the form 12 g r. These two equations together are known as the einstein klein gordon system. The equivalence principle in classical mechanics and quantum. In order to formulate his theory, einstein had to reinterpret fundamental. On wave dark matter, shells in elliptical galaxies, and the. Relativistic wave equations are invariant under lorentz transformations, expressing the invariance of the element of 4vector length, ds 2 dt 2. Geometrical derivation of the kleingordon equation springerlink. The klein gordon equation was proposed by klein and gordon in 1927 but it did not work.
Later, the equation was revived and is now known as klein gordon equation. The hydrodynamic representation of the klein gordon equation kge. While the equivalence principle stands at the very heart of general relativity. Techniques developed for linear and nonlinear wave equations arising from lagrangian theories have been central tools in the dynamical study of einsteins equations.
The klein gordon equation is a wave type of equation for a scalar. Pdf modified general relativity and the kleingordon. To illustrate this point, we solve the problem of a spin zero relativistic particle in a one and threedimensional box using the klein gordon. Klein gordon and dirac equationsin general relativity. Pdf kleingordon equation and wave function for free particle in. You can nd planewave solutions to the klein gordon equation easily. Whatever these constants might be the the equation must satisfy the relativistic energy equation. Rather than trying to force quantum theory on general relativity, or vice versa, this article investigates if a connection between quantum theory and general relativity exists naturally. In the low eld, nonrelativistic limit, the einstein klein gordon system reduces to the poissonschr odinger system. In the general theory of relativity it is well known that the equations. A singularity theorem for einsteinkleingordon theory core. Dirac equations for the motion of a charged particle in a coulomb field do not possess solutions which. A singularity theorem for einsteinkleingordon theory page 5 of 24 121 where the conformal coupling constant. Lecture notes on general relativity columbia university.
From einstein to kleingordon quantum mechanics and relativity aline ribeiro department of mathematics university of toronto march 24, 2002 abstract we study the development from einsteins relativistic energymomentum relation for free particles, e2. Einsteins equations of classical general relativity can be derived from the. Modified general relativity is intrinsically hidden in the klein gordon equation for spins 2 and 32. Mode solutions of the klein gordon equation on a subextremal kerr black hole.
These fields are generally solutions to certain systems of partial differential equations, such as maxwells equations or the klein gordon equation. Deriving the klein gordon equation relativistic quantum. We examine the einstein equation coupled to the klein gordon equation for a complexvalued scalar eld. Conformal kleingordon equations and quasinormal modes. In the general relativity theory, we find klein gordon wave functions in robertsonwalker and schwarzschild spacetime. Kleingordon equation and contain only positive frequencies on m1. Although the form of the equations is no longer manifestly covariant, they are valid for any choice of time coordinate, and for any coordinate system the results are equivalent to those. Relativistic quantum mechanics kleingordon equation dirac. How to use the einstein energymomentum relation to derive the klein gordon wave equation. Separation of variables and exact solution of the klein. Pdf kleingordon equation and wave function in cosmological. Thus, in general, we will have to resort to an approximate procedure. The semilinear kleingordon equation in two and three space.
The klein gordon equation klein fock gordon equation or sometimes klein gordon fock equation is a relativistic wave equation, related to the schrodinger equation. Geometrical derivation of the kleingordon equation. Bergman,introduction to the theory of relativity dover pub. In the scale relativity framework, the schr odinger, klein gordon and dirac equations have been derived from rst principles. Klein gordon scalar field in a gravitational field. So, none of the objections that led to discarding the klein equation held up with time. We examine the einstein equation coupled to the klein gordon equation for a complexvalued scalar field. The new quantum mechanical law shown in equation 42 is more general than schrodinger and klein gordon equations.
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