A quantum gravity tensor equation formally integrating general relativity with quantum mechanics XU DUAN Zhengzhou City Construction Engineering Foundation Institute - 450000 Zhengzhou, People's Republic of China Abstract Extending black-hole entropy to ordinary objects, we propose kinetic entropy tensor, based on which a quantum gravity tensor equation is established. Our investigation results indicate that if N=1 , the quantum gravity tensor equation returns to Schrödinger integral equation. When N becomes sufficiently large, it is equivalent to Einstein field equation. This illustrates formal unification and intrinsic compatibility of general relativity with quantum mechanics. The quantum gravity equation may be utilized to deduce general relativity, special relativity, Newtonian mechanics and quantum mechanics, which has paved the way for unification of theoretical physics. 1. Introduction The unification of general relativity and quantum mechanics, a long-standing unsolved difficult problem, is one of the most important frontier researches in theoretical physics. It has puzzled many famous grandmasters of science for decades, including Albert Einstein. A current viewpoint suggests that they are essentially contradictory in their ideas and explanation of our world. General relativity is a very effective theory concerning macroscopic objects, especially cosmology. But in contrast, quantum mechanics specializes in dealing with molecules and atoms. Both theories have their own scope of application, and seem to be incomplete. It appears that we can’t simply com- bine these two theories to create a single equation of quantum gravity. Although substantial exciting progresses, such as black hole entropy, holographic principle, M (superstring) theory and loop quantum gravity, have emerged in the domain of quantum gravity, yet up till now, all the attempts to discover a theory of formally unified and complete quantum gravity are far from succeeding. Inspired by black-hole thermodynamics, we propose in this paper kinetic entropy tensor and successfully con- struct a quantum gravity tensor equation, which could integrate Schrödinger equation with Einstein field equation self-consistently. This means general relativity and quantum mechanics have intrinsic compatibility, instead of be- ing irreconcilable with each other, as is thought previously. The quantum gravity equation may be utilized to de- duce general relativity, special relativity, Newtonian mechanics and quantum mechanics, which has paved the way for unification of theoretical physics. E-mail: [email protected] - 1 - 2. Derivation of kinetic entropy In 1976 William G. Unruh[1] put forward his famous Unruh effect，indicating that accelerated motion is equivalent to thermal radiation of black bodies. It gives a strong hint that temperature and entropy are closely re- lated to kinetics. However, previously the concept of entropy was only restricted to classical equilibrium state in thermodynamics. To deduce kinetic entropy, we need to take advantage of entropy force together with Unruh for- mula. Entropy force[2,3], as a real force driven by entropy changes of a system, could be written as follows: TdS FdR (1) among which, T and S represent temperature and entropy, respectively. F is entropy force and R is space coordinate. Unruh law[1] in quantum field theory is expressed by 1 a kT (2) 2 c where T is Unruh temperature and a is acceleration. k , and c are respectively Boltzmann constant, re- duced Planck constant and velocity of light in vacuum. By combining (1) with (2), we have: c MRTSc2 kT d = d (3) 2 Furthermore, we acquire: k dS 2 M c2 dR (4) c The integral form of kinetic entropy is: k S 2 MR c2 (5) c 3. Kinetic entropy and Unruh temperature of general objects Black-hole thermodynamics developed during 1970’s reveals that black holes possess temperature and entropy. In fact, general objects (not only black holes) also have kinetic entropy and Unruh temperature. So how to extend thermodynamics of black holes to general objects? First we introduce escape velocity v 2 GM R . It is not hard to find that energy U 、space R and escape velocity v could be generalized into the following relationship, which applies to an ordinary object. UR v2 2 (6) Ep x p 2c - 2 - 3 5 2 xP called Planck scale, xP G c , EP called Planck energy, EP c G , UM c , G representing uni- versal gravitational constant. In appendix (the last section), we will see that (6) could be derived directly from Friedmann equation in flat space-time. Based on (6), kinetic entropy and Unruh temperature of ordinary objects are easy to obtain, including black-hole entropy and black-hole temperature. Specifically speaking, putting (6) into (5), we obtain kinetic entropy of ordinary objects: 2 U R R v2 S 2 k = 2 k (7) 2 Ep xp x p 2c If v c , it’s just Bekenstein’s black-hole entropy[4]. 2 R R2 c 3 S k k (8) BH xp G Furthermore, by transformation of Unruh formula, we get: 2 c v (9) kTR 2 2 2c After (9) is plugged into (6), Unruh temperature of general objects is acquired: 2 2 2 Ep 1 v kT 2 2 (10) 2Mc 2c In particular, when v c , black-hole temperature derived by Hawking [5] is easily got from our method: E 2 c3 kT p = 8 Mc2 8 G M It is not hard to prove that kinetic entropy and Unruh temperature of general objects invariably satisfy the rela- tionship below, which is very useful in the deduction of Newton’s law later. 1 T S Mv2 (11) 2 4. Kinetic entropy tensor Maybe an important enlightenment of general relativity is that theories in the form of tensors are genuinely com- plete. The key to unification of general relativity and quantum mechanics lies in that kinetic entropy must be rewrit- ten as a second-order symmetrical tensor. How to determine the expressions of kinetic entropy tensor Sab ? Obviously by analogy with kinetic entropy (5), we presume kinetic entropy tensor Sab should be defined as: def k S 2 UIX (12) abc ab ab ab - 3 - 4 3 Here UTab ab r , and Tab denotes energy-momentum tensor. Xab stands for tensor of space-time 3 coordinate, and Iab is coefficient matrix. The next crucial step is to write out the specific components of Uab and Xab under Cartesian coordinate system, although they are arbitrary and depend on selection of metric tensor. But we know Uab has a standard form: v vxy v z 1 c c c 2 2 Mc Px c P y c P z c v vv v v v x xx y x z 2 2 2 2 Px c Mv x Mv x v y Mv x v z c c c c UP = Mc2 = Mc 2 abP c Mv v Mv2 Mv v v v v v2 v v ab y y x y y z y x y y y z 2 2 2 2 Pz c Mv z v x Mv z v y Mv z c c c c 2 v v vvy v z v z x z z c c2 c 2 c 2 To satisfy the covariant principle, we designate Xab adopt the same symmetrical form: vxvy v z ct x y z 1 c c c v vxy v z 2 x x x x vx v xvx v y v x v z c c c c c2 c 2 c 2 XPv = ct = ct abvxy v z 2 ab y y y y vy v x v y v y v y v z c c c c c2 c 2 c 2 vv v z zx z y z z v v vv v v2 c c c z x zy z z c c2 c 2 c 2 After repeated speculations and tentative calculations, an important matrix relation is ultimately obtained: v v vxy v z v x y v z 1 1 c c c c c c v 2 v v2v v v v - 0 0 0 v v 2 v v v v x xx y x z2c 2 x x x y x z c c2 c 2 c 2 c c 2 c 2 c 2 0 1 0 0 2 2 vvvvvvyxyyyz vvvvvv yxyyyz 0 0 1 0 c c2 c 2 c 2 c c 2 c 2 c 2 0 0 0 1 v v2 v v 2 vz v x v zy z v z v z v x v z y z v z c c2 c 2 c 2 c c 2 c 2 c 2 v vxy v z 1 c c c 2 v v vx v xx y v x v z v 2 c c2 c 2 c 2 = 2 2 2c vy v x v y v y v y v z 2 2 2 c c c c v v 2 vz v x v zy z v z 2 2 2 c c c c So we learn proper choice of coefficient matrix Iab helps to make Sab , Uab and Xab always covariant. Hence coefficient matrix Iab is determined as follows: v 2 PPPabI ab ab = 2 ab 2c - 4 - Or v2 -2 0 0 0 2c Iab 0 1 0 0 0 0 1 0 0 0 0 1 4 4 If we employ Bits number (or degree of freedom) tensor Nab to fulfill the quantization of entropy: Sab N ab k then there is: 2UIXNab ab ab = ab c (13) We have actually established a half of the quantum gravity equation. 5. Wave function tensor and kinetic entropy tensor Is there any correlation between kinetic entropy and wave function? Our investigation offers a definite answer.