PHYSICAL REVIEW D 86, 124017 (2012) Higher order equations of motion and gravity Claus La¨mmerzahl1,2,* and Patricia Rademaker1,3,† 1ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany 2Institute for Physics, University Oldenburg, 26111 Oldenburg, Germany 3Institute for Theoretical Physics, Leibniz University Hannover, Appelstrasse 2, 30167 Hannover, Germany (Received 1 September 2010; published 6 December 2012) Standard fundamental equations of motion for point particles are of second order in the time derivative. Here we are exploring the consequences of fundamental equations of motion with an additional small even higher order term to the standard formulation. This is related to two issues: (i) higher order equations of motion will have influence on the definition of the structure of possible interactions and in particular of the gravitational interaction, and (ii) such equations of motion provide a framework to test the validity of Newton’s second law which is the basis for the definition of forces but which assumes from the very beginning that the fundamental equations of motion are of second order. We will show that starting with our generalized equations of motions it is possible to introduce the space-time metric describing the gravitational interaction by means of a standard gauge principle. Another main result within our model of even higher order derivatives is that for slowly varying and smooth fields the higher order derivatives either lead to runaway solutions or induces a zitterbewegung. We confront this higher order scheme with experimental data. DOI: 10.1103/PhysRevD.86.124017 PACS numbers: 04.20.Cv, 04.20.Fy I. INTRODUCTION the basic structure of the equation of motion. If, for ex- ample, the equation of motion is of higher than second The most basic approach to the mathematical descrip- order in the time derivative, then interactions could be tion of nature is provided by Newton axioms [1]. The most introduced in a different way and, thus, can have a different important of them state that (i) there are inertial systems, structure as we will show below. As a consequence, it is (ii) the acceleration of a body with respect to an inertial system is given by very important (i) to explore the structure of interactions and in particular of the gravitational interaction in the case mx€ ¼ Fðx; x_ ;tÞ; (1) of higher order fundamental equations of motion, and where m is the (inertial) mass of the body, x€ its accelera- (ii) to develop a theoretical framework for the description tion, and F the force acting on the body which may depend of experiments testing the order of the equations of motion. on the position and the velocity of the particle, and that (iii) Equations of motion of higher order are related to a actio equals reactio [2]. different initial value problem: one needs more initial Leaving aside the fundamental and still unresolved prob- values beyond the initial position and initial velocity. In lem of how to really define an inertial system (see, e.g., physical terms this means that with respect to the time Ref. [3]), the second Newton axiom is a tool to explore the coordinate the equation of motion is more nonlocal than forces and, thus, to measure the fields acting on particles. The the ordinary second order equations. The extreme case that electromagnetic field, for example, can be explored and (formally) an infinite number of initial values are needed is measured through the observation of the acceleration of related to equations of motion with memory as, e.g., gen- charged particles under different conditions (different initial eralized Langevin equations where theR force equation pos- t ð À 0Þ ð 0Þ 0 conditions, different charges, etc.). Thus, by means of the sesses an additional term of the form 0 C t t x_ t dt , dynamics of the form (1) the electromagnetic interaction and, see, e.g., Ref. [4]. in principle, all other interactions are defined. In many cases Also within quantum gravity scenarios it might be one uses quantum equations of motion like the Schro¨dinger expected that the effective equation of point particles or Dirac equation which, via the path integral approach or the may contain small higher order time derivatives. In fact, Ehrenfest theorem, for example, are also based on (1). All since space-time fluctuations in the sense of, e.g., fluctua- these considerations extend to relativistic equations of mo- tions of the metrical tensor, yield Langevin-like equations tion. And all this is also the basis of the introduction of of motion also quantum gravity scenarios naturally are interactions through a gauge formalism which is basic for expected to lead to effective higher order equations of the theoretical description of the standard model. motion where the additional higher derivatives probably From this observation it is clear that what one defines as scale with, e.g., the Planck length. interaction or as the corresponding force field depends on Higher order derivatives in equations of motion occur in effective equations which take backreaction into account. *[email protected] One example for that is the Abraham-Lorentz equation for †[email protected] charged particles taking into account the electromagnetic 1550-7998=2012=86(12)=124017(10) 124017-1 Ó 2012 American Physical Society CLAUS LA¨ MMERZAHL AND PATRICIA RADEMAKER PHYSICAL REVIEW D 86, 124017 (2012) waves radiated away [5] or the radiation damping equation d 0 ¼ rL À r L: (3) in gravity where the emitted gravitational waves are taken dt x_ into account, [6]. In the electromagnetic case the leading term is a third time derivative which leads to unphysical We obtain the same equation of motion from two runaway solutions which still is an unresolved problem. Lagrange functions if they differ by a total time derivative However, these equations discussed in relation with radia- only, tion damping are no fundamental equations, they are ef- d Lðt;x;x_ Þ!L0ðt;x;x_ Þ¼L ðt;x;x_ Þþ fðt;xÞ; (4) fective equations emerging from the fact that one no longer 0 dt regards the particles as test particles. Here we are only interested in the fundamental equations of motion. where the function f is allowed to depend on t and x only. Since spin is some element of nonlocality it is not aston- We can expand the total time derivative, ishing that also the dynamics of spinning particles effectively 0ð x xÞ¼ ð x xÞþ ð xÞþxÁr ð xÞ can be described by means of a higher order theory [7]. L t; ; _ L0 t; ; _ @tf t; _ f t; : (5) Pais and Uhlenbeck [8] studied higher order mechanical models as a toy model for discussing properties of higher We are now invoking the gauge principle which pre- order field theories where higher order derivatives came in scribes the replacement, naturally in noncommutative models [9] or are introduced ð xÞ! ð xÞ r ! Að xÞ in order to eliminate divergences; see, e.g., Ref. [10]. @tf t; q t; ; f q t; ; (6) Recently, it has been shown in Ref. [11] that the energy of the Pais-Uhlenbeck oscillator is bounded from below, is where q is the coupling parameter (charge). The new ð xÞ Að xÞ unitary and is free of ghosts; see also Ref. [12] for further functions t; and t; are the scalar and vector studies in this direction. potential of the Maxwell theory. Then the Lagrangian In the following we will consider fundamental equations coupled to these potentials reads of motion of even higher order which can be derived from a 0ð x xÞ¼ ð x xÞ ð xÞþ xÁAð xÞ variational principle. In order to be able to confront these L t; ; _ L0 t; ; _ q t; q _ t; : (7) modified equations of motion with experimental data, one ð x xÞ¼1 x2 first has to investigate the structure of interactions. This With the choice L0 t; ; _ 2 m _ we obtain the standard will be done by using a gauge principle for a second order Lorentz force equation of a charged particle moving in an Lagrange formalism (which in principle can be applied electromagnetic field. to Lagrangians of all orders). The solutions of the equa- Now we generalize this approach to higher order tions of motion coupled to these generalized gauge fields Lagrange functions. give—according to the chosen sign of the additional even higher derivative term—only runaway solutions or a B. Second order formalism zitterbewegung showing that in the latter case the standard In the second order formalism we consider Lagrange equations of motion are rather robust against addition of functions of the form higher order terms. We also discuss the experimental pos- ¼ ð x x xÞ sibilities to search for effects related to such higher order L L0 t; ; _ ; € (8) time derivatives. As an interesting by-product of the cor- responding higher order gauge formalism we obtain the from which we obtain the fourth order equation of standard space-time metric as an ordinary gauge field. motion, d d2 0 ¼ rL À r L þ r L II. LAGRANGE FORMALISM x_ 2 x€ dt dt d d First we will use the Lagrange formalism in order to ¼ rL À r L À r L ; (9) introduce interactions into a theory containing higher order dt x_ dt x€ derivatives. We introduce interactions by means of a gauge r principle. We set up our notation by repeating shortly the where a denotes the gradient with respect to the variable standard first order formalism and then apply the gauge a. Also in this case we obtain the same equation of motion principle to a second order Lagrange formalism.