Non-linear energy conservation theorem in the framework of Special Relativity
aa r X i v : . [ g r- q c ] M a y Non-linear energy conservation theorem in the framework of Special Relativity
Gin´es R.P´erez Teruel Departamento de F´ısica Te´orica, Universidad de Valencia, Burjassot-46100, Valencia, Spain
Abstract
In this work we revisit the study of the gravitational interaction in the context of the Special The-ory of Relativity. It is found that, as long as the equivalence principle is respected, a relativisticnon-linear energy conservation theorem arises in a natural way. We interpret that this non-linearconservation law stresses the non-linear character of the gravitational interaction.The theorem re-produces the energy conservation theorem of Newtonian mechanics in the corresponding low energylimit, but also allows to derive some standard results of post-Newtonian gravity, such as the formulaof the gravitational redshift. Guided by this conservation law, we develop a Lagrangian formalismfor a particle in a gravitational field. We realize that the Lagrangian can be written in an explicitcovariant fashion, and turns out to be the geodesic Lagrangian of a curved Lorentzian manifold.Therefore, any attempt to describe gravity within the Special Theory, leads outside their own do-mains towards a curved space-time. Thus, the pedagogical content of the paper may be useful asa starting point to discuss the problem of Gravitation in the context of the Special Theory, as apreliminary step before introducing General Relativity.
I. INTRODUCTION
The General Theory of Relativity (GR) is the mostaccepted theory nowadays to describe the behaviour ofthe classical gravitational field. The theory is probablyone of the most well-tested theories in physics. Ingeneral, there has been an excellent level of agreementbetween theory and experiments in scales that rangefrom millimeters to astronomical units, scales in whichweak and strong field phenomena can be observed[1]. Due to their accurate predictions, it is generallyaccepted that the theory should also work at larger andshorter scales, and at weaker and stronger regimes.Nevertheless, there are serious and fundamental openproblems that remain to be solved. For instance, to ex-plain the rotation curves of spiral galaxies, we must ac-cept the existence of vast amounts of unseen matter sur-rounding those galaxies. A similar situation occurs withthe analysis of the light emitted by distant type-Ia super-novae and some properties of the distribution of matterand radiation at large scales. To make sense of thoseobservations within the framework of GR, we must ac-cept the existence of yet another source of energy withrepulsive gravitational properties[2]. In addition, theoutstanding difficulties to harmonize the conceptual andmathematical framework of GR with the rest of physics,in particular the problem of consistently combining GRwith Quantum Mechanics (QM), have risen the inter-est of the physics community in the search for modifiedtheories of gravity. These modified theories of gravity in-clude a wide variety of different approximations: Mondtheories [4][5], scalar-tensor theories[3], f ( R ) generaliza-tions in metric and Palatini formalism respectively [6–8],and even studies about the implications of a violation ofthe weak and strong equivalence principle. (see Ref.[9]for a recent one). Due to the fact that the rest of theclassical and quantum field theories are formulated in a flat spacetime, a natural possibility seems the seek fora relativistic theory of gravitation purely constructed inMinkowski spacetime, i.e, a theory of gravity subjectedonly to the constraints and principles of Special Rela-tivity (SR). Indeed, this approach was historically thefirst considered by Einstein himself, although he eventu-ally abandoned it to pursue a theory constructed withthe aid of Riemannian geometry. The reasons to un-derstand the rejection of such a promising approach arecomplex and sometimes not well explained in the intro-ductory lectures of GR, where the theory is presented asa sort of revelation. Even in our days, we still see theo-retical attempts to search for a theory of gravitation inMinkowski’s spacetime, with the hope that such achieve-ment will make more factible the unification of gravitywith the rest of physics. Then, why is this a failed re-search program? Why is not possible to find a satis-factory special-relativistic theory of gravitation withoutcrossing the own domains of SR?In this work, we revisit the problem of gravitation inSR with open mind , and find that if we assume thatthe equivalence principle is of universal validity, then itcan be derived a non-linear conservation law that relatesthe variation of the relativistic mass with the variation ofthe gravitational potential. This formula generalizes ina natural way the energy conservation theorem of New-tonian Mechanics. Furthermore, the Lagrangian formal-ism that we obtain for the model possesses an intrinsicgometric meaning: it can be rewritten as the geodesicLagrangian of a Lorentzian manifold. Then, any con-sistent attempt to describe gravity in the framework ofSR, will lead outside the domains of this theory. Weconclude that our approach, due to their simplicity, maybe employed in a pedagogical manner to introduce theproblem of Gravitation in SR as a previous step beforeaddressing the study of GR, and it can provide a tool tounderstand why gravity is different from the other forcesof nature: Gravitation cannot be described by any flatpace-time-based theory. II. THE EQUIVALENCE PRINCIPLE AND THENON-LINEAR ENERGY CONSERVATIONTHEOREM
Let us consider the movement of a test particle in anexternal gravitation field in the framework of SR. We as-sume that the four-momentum p µ , of the test particle inan arbitrary reference frame is given by ( E, c~p ). Any dif-ferential variation of their energy due to their movementcan be expressed as dE = dm i c (1)where dm i denotes the differential variation of the iner-tial relativistic mass of the test particle. On the otherhand, since the particle is subjected to a conservativeforce that derives from a potential, the infinitesimal workmade by the gravitational field on the particle will be dW = − m g ∇ φ · dr = − m g dφ (2)Where m g is the gravitational mass of the particle.Equaling both equations, we get dm i c = − m g dφ (3)The acceptance of the equivalence principle implies thatthe massess that appear in both sides of the last equationare strictly equal. In these conditions, if m i = m g we canrearrange terms to write Z r r dmm = − Z r r dφc (4)The integration of this equation between two arbitrarypoints provides, m ( φ ) e φ /c = m ( φ ) e φ /c (5)Or equivalently, m p − β ( φ ) exp( φ/c ) = C (6)Where C is a constant of motion, and m is a constantcharacteristic of the particle which does not depend on φ . What is the physical meaning of this formula? Itrepresents a non-linear generalization of the energy con-servation theorem of newtonian mechanics. Note thatthis conservation law emerges automatically as long asthe equivalence principle is accepted. In plain words,equation (5) is expressing the following: when the parti-cle moves far from the sources of the gravitational field,the value of the potential φ increases, and therefore theirrelativistic mass, m ( φ ) = m √ − β ( φ ) should decrease be-cause the factor, β decreases. Note that the oppositesituation takes place when the particle approximates to the sources of the gravitational field: in this case the po-tential φ decreases (becomes more negative), and thenthe relativistic mass increases due to the increase of thevelocity of the particle. We should mention that the ex-ponential mathematical form of the function m ( φ ) hasbeen obtained elsewhere (see for instance [10],[11]) al-though the presence of a general conservation law is notrecognized in these works, nor their full implications. In-deed, we will show in the next sections that guided bythis conservation theorem it is possible to formulate aLagrangian theory that can be interpreted in geomet-rical terms as the geodesic Lagrangian of a Lorentzianmanifold. A. The Newtonian limit
In order to prove the robustness of this result, let usconsider the case of a particle initially placed at rest at apoint r from the origin of a coordinate system. Supposethat the measured value of the potential at r is φ . If theparticle moves to another point r where the potentialis φ , under the effect of the gravitational field, theirrelativistic mass at r according to Eqs. (5-6) will be: m ( φ ) = m e ( φ − φ ) /c (7)Then, according to STR the energy of the particle mea-sured by the same observer at the point r will be E = m ( φ ) c = m c e ( φ − φ ) /c = T + m c (8)where T is the kinetic energy acquired by the particledue to the value of their velocity at r . Note that ifthe variation, φ − φ is small compared with c , we canapproximate Eq. (8) as follows E = m c e ( φ − φ ) /c = m c (cid:16) φ − φ ) /c + ... (cid:17) = m c + m ( φ − φ ) = T + m c (9)This implies,∆ T = T − T = m ( φ − φ ) = − ∆ U (10)This result recovers in a natural way the conservationof the mechanical energy of newtonian mechanics. In-deed, note that the particle was initially at rest, ( T = 0)therefore Eq.(10) express that the increase of the kineticenergy ∆ T of the particle is equal to the decrease of theirpotential energy. On the other hand, using the generalenergy-momentum relation E = c p + m c , we canfind a closed formula for the momentum of the particleat the point r in terms of the variation of the potentialbetween both points as follows p = 1 c q E − m c = m c p e φ − φ ) /c − φ − φ ) /c <<
1, the approximation of the exponential upto the second order provides: p ≃ m c p φ − φ ) /c − m c p φ − φ ) /c = p m T (12)In total agreement with the result of non-relativistic clas-sical mechanics, i.e, T = p / m B. The linear energy conservation theorem ofclassical mechanics as a particular case
We can provide a more transparent proof of how thenon-linear energy conservation theorem (5), reducesto the linear conservation of energy of Newtonianmechanics, E = T + U in the appropriate limit. Indeed,note that (5), can be written in the form m p − β e φ /c = m p − β e φ /c (13)For small velocities compared with the speed of light, β = v /c <<
1, and weak gravitational fields, φ/c <<
1, the last equation can be approximated as m (cid:16) β (cid:17)(cid:16) φ c (cid:17) = m (cid:16) β (cid:17)(cid:16) φ c (cid:17) (14)where we have neglected the contributions that go as O (1 /c ). After a direct computation, the last equationprovides the result:12 m v + m φ + 12 m v φ c = 12 m v + m φ + 12 m v φ c (15)As we can see, this is the conservation theorem ofthe mechanical energy of classical physics, T + U = C ,corrected by a first non-linear contribution given by, m v φ/ c . It is interesting to note that the presenceof a non-linear energy conservation theorem stresses thenon-linear character of the gravitational interaction. Wewill show in the next section that the full non-linear con-servation law (5-6) can be viewed as a conserved canon-ical Hamiltonian. C. Lagrangian formulation and equations of motion
We postulate the following non-linear relativistic La-grangian for a particle in a gravitational field: L = − m c p − β exp( φ/c ) (16) In the non-relativistic limit this Lagrangian becomes: L ≈ − m c + 12 m v − m φ + 12 m φ v c + ... ≈ T − U − m c (17)Then, this Lagrangian possesses an acceptable low en-ergy behavior. Now, in order to proceed further, we as-sume that the relation between the Lagrangian L , andthe Hamiltonian H is the standard, i.e, they are relatedthrough a Legendre transform, namely: H = X i p i v i − L (18)Where the canonical momenta p i , are given by, p i = ∂ L ∂v i = m v i p − β exp( φ/c ) (19)And we have assumed that the potential φ ( r ) is onlya function of r and does not depend on the velocities.Substituing this result in (18) we obtain H = (cid:16) X i m v i p − β + m c p − β (cid:17) exp( φ/c )= (cid:16) P i m v i + m c (1 − β ) p − β (cid:17) exp( φ/c )= m c p − β exp( φ/c ) (20)Therefore, as we suspected, the non-linear conservationlaw (5,6), is nothing but the conservation of the canonicalHamiltonian (20), times a constant. On the other hand,the equations of motion for the Kepler problem can beobtained by application of the Euler-Lagrange equationsto the Lagrangian (16). These equations for the case ofmovement in two dimensions will be: ∂ L ∂r − ddt ∂ L ∂ ˙ r = 0 (21) ∂ L ∂θ − ddt ∂ L ∂ ˙ θ = 0 (22)Since, β = v /c = ( ˙ r + r ˙ θ ) /c , the Lagrangian func-tion (16) will acquire the following form L = − m c s − ˙ r + r ˙ θ c exp( φ ( r ) /c ) (23)Then, the coordinate θ is cyclic, which implies that aconserved constant of motion is present. Indeed, apply-ing (22) we obtain ddt (cid:16) m r ˙ θ p − β exp( φ ( r ) /c ) (cid:17) = ddt ( H r ˙ θ ) = 0 (24)Since we know that, H = exp( φ/c ) m / p − β is con-stant given the previous results, it is easy to see thatequation (24) is expressing the conservation of the an-gular momentum, r ˙ θ = C . Regarding the other Euler-Lagrange equation, with a bit of algebra we find: (cid:16) − ∂φ∂r m p − β + l m r p − β (cid:17) e φ/c = ddt (cid:16) m ˙ r p − β e φ/c (cid:17) (25) Where, l = m r ˙ θ . According to these equations of mo-tion, the product m · exp( φ/c ), determines the inertiaof a mass point. In other words: the bigger the potentialis, the bigger will be the resistence exerted by the parti-cle in response to a variation in its velocity. Developingthe derivative of the right hand side, we can remove thefactor e φ/c from the analysis. Doing this, we obtain thecompact form of the equation of motion for the Keplerproblem as: ddt (cid:16) m ˙ r p − β (cid:17) = − ∂φ∂r m p − β + l m r p − β − m ˙ rc p − β dφdt (26)The right hand side can be seen as the effective force thatfeels the particle. When β <<
1, we naturally recoverthe equation of motion of Newtonian mechanics. How-ever, the situation is more involved when high velocitiesare considered, because the term of the right hand sidethat includes the gradient decreases (multiplies the factor p − β ), while the other two increase with the veloc-ity. This effect may have interesting consequences, forexample, to compute the prediction of this model for theanomalous perihelion precession of planetary orbits; In-deed, the comparison between this prediction with thoseof GR, will falsify the model, or at least will fix strongconstraints on the structure of the family of possible non-linear Lagrangians. On the other hand, the non-linearLagrangian (16), possesses a deep geometrical meaning.Indeed, note that it can be rewritten as: L = − m c p − β e φ/c = − m c p exp(2 φ/c )(1 − β )= − m cdt q g µν ( x ) dx µ dx ν = − m c dsdt (27)Where, g µν ( x ) = exp(2 φ/c ) · η µν . This is the geodesicLagrangian of a curved manifold! Then, if we try to formulate the gravitational interaction within a flatspace-time, the consistency of the theory will push youout of the Special Theory towards a curved space-time.Therefore, the Special Theory of Relativity cannotcontain by itself the gravitaional interaction. This canbe useful to explain why gravitation is different fromthe other fundamental forces of nature. III. SUMMARY AND CONCLUSIONS
In this work, we have revisited the problem of themovement of a particle in a gravitational field in the con-text of SR. We have found that the equality of inertialand gravitational massess allows to derive a non-linearenergy conservation law that generalizes the energy con-servation theorem of Newtonian mechanics. This non-linear conservation law has provided a hint of how toconstruct a consistent Lagrangian formalism. As we ex-pected, the Lagrangian function turns out to be non-linear, which via the corresponding Euler-Lagrange equa-tions, provides equations of motion that are also non-linear, but with a correct classical limit. In addition,we have realized that the Lagrangian function possessesa deep geometrical insight. In fact, it can be rewrittenas the covariant geodesic Lagrangian of a curved mani-fold. Then, starting from the Special Theory perspective,the consistency of the theory implies a natural transitionfrom a flat space-time to a curved space-time. This canjustify the pedagogical use of this model to illustrate abeautiful transition between SR and GR. On the otherhand, it would be very interesting for future works thestudy of possible generalizations of this conservation lawfor velocity-dependent potentials of Berger’s type [12],or even for more general retarded potentials[13], whichhave been shown to reproduce the anomalous precessionof Mercury’s perihelion.
IV. ACKNOWLEDGEMENTS
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