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Home , Field equation, General relativity, Gravity, Quantum gravity

A quantum equation formally integrating with quantum

XU DUAN Zhengzhou City Construction Foundation Institute - 450000 Zhengzhou, People's Republic of China

Abstract Extending -hole to ordinary objects, we propose kinetic entropy tensor, based on which a 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 equation. This illustrates formal and intrinsic compatibility of general relativity with . The quantum gravity equation may be utilized to deduce general relativity, , Newtonian mechanics and quantum mechanics, which has paved the way for unification of theoretical .

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 . It has puzzled many famous grandmasters of for decades, including . A current viewpoint suggests that they are essentially contradictory in their ideas and explanation of our . General relativity is a very effective concerning macroscopic objects, especially . But in contrast, quantum mechanics specializes in dealing with molecules and atoms. Both 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 entropy, , M (superstring) theory and , 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 , we propose in this paper kinetic entropy tensor and successfully con- struct a quantum gravity tensor equation, which could integrate Schrödinger equation with Einstein 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]

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2. Derivation of kinetic entropy

In 1976 William G. Unruh[1] put forward his famous ，indicating that accelerated is equivalent to thermal of black bodies. It gives a strong hint that temperature and entropy are closely re- lated to . 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 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 coordinate. Unruh law[1] in is expressed by 1 a kT  (2) 2 c where T is Unruh temperature and a is . k ,  and c are respectively Boltzmann constant, re- duced Planck constant and of in . 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 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 

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3 5 2 xP called Planck scale, xP  G c , EP called Planck energy, EP   c G , UM c , G representing uni- versal . In appendix (the last section), we will see that (6) could be derived directly from Friedmann equation in flat space-. 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) kTR  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 ’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 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 -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) abc ab ab ab

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4 3  Here UTab ab  r  , and Tab denotes energy- tensor. Xab stands for tensor of space-time 3  coordinate, and Iab is coefficient . The next crucial step is to write out the specific components of Uab and

Xab under Cartesian , although they are arbitrary and depend on selection of tensor. But we know Uab has a standard form:

v vxy v z  1  c c c  Mc2 P c P c P c  2 v v x y z vx v xx y v x v 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  abP 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 XPv  = ct   = ct  abvxy 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 zx 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 z2c 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:

2  v PPPabI ab  ab = 2   ab 2c 

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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 of entropy:

Sab N ab k then there is:

2UIXNab ab  ab = ab  c (13) We have actually established a half of the quantum gravity equation.

5. function tensor and kinetic entropy tensor Is there any correlation between kinetic entropy and ? Our investigation offers a definite answer. Seeing that kinetic entropy is quantized, we have: k S  2 MN c2  r =   k （ N is Bits number or degree of freedom） c

Mc2  r N  2 c Supposing a macroscopic body is made up of N microscopic . M means of a macroscopic body, and m refers to mass of a microscopic . It is obvious M= N  m . In addition, wave function obeys Schrödinger equation: m r2   0 exp    2i t  Thus

2   N m r2M c 2 v -2N  iln   2   = 2   r  0  t  c c v = N c This implies Bits number (or degree of freedom) N and particle number N should satisfy the relation:

2 v    N  = -2 N  iln  (14) c   0  In order to fit kinetic entropy tensor and Bits number tensor, we have to extend wave function to a second-order symmetrical tensor  ab . When we define wave function tensor as:

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vxvy v z  1  c c c  2 v v vx v xx y v x v z  def   c c2 c 2 c 2    ln ab  ln   2  0  vy v x v y v y v y v z  2 2 2  c c c c  2 vz v x v zvy v z v z  2 2 2  c c c c  4 4 we discover there exists simple correspondence between wave function tensor  ab and Bits number tensor Nab via a series of matrix calculations:

Nab = -2 N  iln  ab Detailed calculations are given as following: 2UIX  N = ab ab ab ab c vv v   v v v  1xy z 1 x y z c c c   c c c    2    2v 2 v vvx v y v v  -2 0 0 0   v v v x v y v v  x x x z2c x x x z Mc2 ct c c2 c 2 c 2     c c 2 c 2 c2  =2    0 1 0 0    E x v v v v2 v v  v v v v2 v v  p p y x y y y z 0 0 1 0  y x y y y z 2 2 2  2 2 2  c c c c   c c c c   0 0 0 1    v v 2   v v 2 vz v x v zy z v z  vz v x v zy z v z  c c2 c 2 c 2  c c2 c 2 c 2 

vxvy v z  vxvy v z  1  1  c c c  c c c  2 2 v vvx v y v v  v vvx v y v v  x x x z 2 x x x z MRc2 R vc c2 c 2 c 2    v 3 c c2 c 2 c 2  =2   =2       E x 2c2  x  4c3 2 p pvy v x v y v y v y v z   p  vy v x v y v y v y v z  2 2 2  2 2 2  c c c c  c c c c  v v 2 v v 2 vz v x v zy z v z  vz v x v zy z v z  c c2 c 2 c 2  c c2 c 2 c 2   vv v  1 xy z  c c c    2 v v vx v xx y v x v z  1 v  c c2 c 2 c 2      = N    2 2 c  vy v x v y v y v y v z   2 2 2   c c c c  2 vz v x v zvy v z v z   c c2 c 2 c 2  

= -2 N  iln ab This leads to a complete quantum gravity tensor equation:

2UIXNab ab  ab = ab c = -2   N  c  iln  ab (15)

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6. Discussions about quantum gravity equation In this section we will focus on particle number N . Two special cases will be discussed in terms of N value. When N=1, (corresponding to a single free particle), quantum gravity equation (15) becomes:

vxvy v z   v x v y v z  1   1  c c c c c c   v2    v v2v v v v  - 0 0 0  v v 2 v v v v  x xx y x z2c2  x x x y x z c c2 c 2 c 2     c c 2 c 2 c 2  mc2   0 1 0 0   ct    vvvvvv2   vvvvvv 2  yxyyyz0 0 1 0  yxyyyz 2 2 2   2 2 2  c c c c  c c c c   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 vz  c c2 c 2 c 2   c c 2 c2 c 2  v vxy v z  1  c c c  2 vx v xvx v y v x v z    c c2 c 2 c 2     =-i c ln   2  0  vy v x v y v y v y v z  2 2 2  c c c c  2 vz v x v zvy v z v z  c c2 c 2 c 2  By detailed matrix calculations, Schrödinger equation of a single free particle emerges exactly in integral form:

 1 2 px x+p y  y+p z  z- t  -i ln   mv  0 2

Of course, here we assume at initial time t0 =0, the particle is located at the origin of coordinates (initial coordi- nate r0 =0). It further leads to the inference:

m r 2    0 exp     2i t  That is to say, Schrödinger equation in integral form could be derived from quantum gravity equation (15) with N=1.

While N is sufficiently large (for example, N 1023 ), our research subject is converted to a macroscopic ob- ject containing enough microscopic particles. Here three basic principles must be satisfied simultaneously: (1) The macroscopic object obeys Newtonian mechanics:

2GM v  R (2) Every microscopic particle within the macroscopic object obeys Schrödinger equation:

m r 2    0 exp     2i t  (3) Holographic principle

The following calculation manifests that Bits number tensor Nab can only equal to:

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3 XXab ab v NIab= 2   ab   3 xp x p 2c Details are given below: v vxy v z  1  c c c  2 vx v xvx v y v x v z   c c2 c 2 c 2    Nab= -2 N  i ln  ab = -2   N  i ln  2  0 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  c c2 c 2 c 2   = -2  N  iln  Pab  0

m r2 M r 2 2GM r2 v r 2 v 3 = 2 N   PP = 2      2    PP=2   2 tab 2  t ab r 4Gab 4G  ab 2 2 r  v3  ct  v 3 = 2   PPIP = 2          3ab   3 ab ab ab xp  4c  x p  2c 3 XXab ab v = 2  Iab   3 xp x p 2c It means that quantum gravity equation (15) with N sufficiently large is practically equivalent to: UXXX 3 ab ab ab ab v 2 INIab  = ab = 2    ab   3 Ep x p x p x p 2c Quite evidently, we have:

3 UXab ab v   3  Ep x p 2c  Provided both sides of the above are divided by V  4 3 r3 , we obtain at once:

vxvy v z   v x v y v z  1   1  c c c   c c c  2v v 2 v v vx v xx y v x v z   v x v x x y v x v z  2 2 2 2   2 2 2  8G2 c c  3 v  ccc c c c  4c2 = 2 2   2 cvvvvvvyxyyyz  r c   vvvvvv yxyyyz  2 2 2   2 2 2  c c   ccc c c c  2 2 v v vvy v z v   v v v v y v z v  z x z z   z x z z  c2 2 c 2   ccc c 2 c 2 c 2  According to appendix, this is precisely Einstein field equation under RW metric in flat space-time: 8 G T  G c 4 ab ab

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Interestingly, for an object, when it contains only one free particle ( N=1), it is subject to Schrödinger equation. When it includes enough microscopic particles (we call it a macroscopic body), thanks to domination of holograph- ic principle[2], it conforms to general relativity. This explains the reason why Einstein field equation only applies to macroscopic bodies, instead of microscopic particles. It should be stressed that theoretical mistakenly believe there exists incompatibility between general relativity and quantum mechanics. But our research demonstrates that at least under RW metric, Einstein field equ- ation and Schrödinger equation could be unified. Besides, our perspective is quite different from superstring and loop quantum gravity. Now let’s return again to kinetic entropy (5) and Unruh formula (2) to reproduce Newton’s law. Referring to Verlinde’s paper[2], just putting (5) and (2) into (11), we get Newton’s second law F = Ma immediately. In addition, if we put (7) into (11), with Unruh formula (2), Newton’s law of universal is also suc- cessfully duplicated: GM GMM a = and F = M a = R2 R2

7. The generalized quantum gravity equation The quantum gravity equation (15) discussed in the 6th section is based on RW metric in flat space-time. We should generalize (15) so that it is tenable for arbitrary metrics in -time. We have known, under RW metric in flat space-time,

3v 2  and X = ct  P Gab = 2 2   P ab ab  ab r c  So

1 3 XG=   ct  ab ab 3 Plus 4  3 UTab= ab   r  3  Substitution of them into (15), the generalized quantum gravity equation is obtained:

4 3 N c r3  ct  TIG   =ab = -N c  iln  (16) 9ab ab ab 2 ab

in which Gab means , and Tab means energy-momentum tensor. Coefficient matrix Iab must satisfy:

 2  TTTabI ab  ab = v   ab 2  Considering N=1, under RW metric in flat space-time, ( = 0 ), as long as we write down:

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2 Tab = c  P ab

3v 2 

Gab = 2 2   P ab r c    ln ab ln   P ab  0 

v 2 

PPPabI ab  ab = 2   ab 2c  the direct inference of Schrödinger equation will be reproduced:

m r 2    0 exp     2i t  It manifests that generalized quantum gravity equation (16) might return to Schrödinger equation spontaneously under RW metric in flat space-time. Thereby we can draw a conclusion that Schrödinger equation holds for flat space-time background which is isotropic and even. While N is sufficiently large, holographic principle begins to play a leading role. For arbitrary metrics, holographic principle takes the form of:

3 XXab ab v NIab= 2   ab   3 xp x p 2c

 3 3 GGab ab = r ct   Iab  9 xp x p If we plug it into generalized quantum gravity equation (16), Einstein field equation arises immediately: 8 G T  G c 4 ab ab It signifies, by means of holographic principle, equation (16) could accurately return to Einstein field equation. In fact, it is holographic principle that makes probability give way to certainty for macroscopic bodies.

8. Special relativity and kinetic entropy Finally we tentatively deduce special relativity only from kinetic entropy. The evident superiority of kinetic en- tropy lies in its simple and convenient deduction of special relativity, without use of and the hypothesis of ‘constancy of light velocity’. We know (7) gives:

2 R  v2 S   k      2 x p  c

If we regard black hole entropy as the maximum equilibrium entropy S0 , which equals

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2 R  S  k 0   xp  then non-equilibrium entropy S can be represented as:

v2  SSS0  S  0  1  2  c  In practice we have got Lorentz relationship of entropy with velocity. It is amazing that other physical quantities varying with velocity are all derived from it very easily. (1)For instance, we derive the relation of space interval R with velocity. As we know:

2 R  v2  S  2 k     2  x p  2c  Meanwhile, covariance requires:

2 R  v2  S  2 k 0  0   2  x p  2c  We conclude immediately:

v2 RR  1- 0 c2 (2)Next, we attempt to derive the relation of particle number N with velocity. Quantization of entropy is taken into consideration: S  Nk Of course, covariance requires:

S0 N 0 k Thus Lorentz relationship of N with velocity is easily got:

v2  NN  0  1- 2  c  (3)The following equations are needed to develop the relation of macroscopic mass M with velocity:

Mc2 R v 2   2  Ep x p 2c 

2 2 M0 c R 0 v   2  Ep x p 2c  Obviously we could reach a conclusion:

v2 M M 1- 0 c2 (4)The relation of microscopic mass m with velocity is easily derived on the of:

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M N m

M0 N 0 m 0 Synthesizing the above conclusions, we have: m m  0 v2 1- c2 (5)We further explore the relation of temperature T with velocity from : M 2TdN S

M 2 T dN 0 0 0 S It’s not hard to infer: T T  0 v2 1- c2 Maybe Lorentz transformation and principle of constancy of light velocity are both unnecessary, only kinetic en- tropy will suffice.

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Appendix

RW metric and matrix components of Einstein field equation First of all, two rational hypotheses are given: (1) Space time is inherently characterized by cosmological prin- ciple and Robertson-Walker metric[6], which suggest space is isotropic and is evenly-distributed. (2) Space is flat, i.e. constant in RW metric   0 . So RW metric is simplified into:

2 2 2 2 2 2 2 2 2 ds -c dt +R（）（ t dr +r d +sin  d  ）  Complicated calculations show that under RW metric with  = 0 , the nonzero components of Ricci tensor are: (here space coordinate is converted into r , so as to distinguish it from Ricci ) a R -3 00 rc2

1 a v2  Rab -2 +2 2  g ab （ a,b=1,2,3） cr r 

Ricci scalar is

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6 a v2  R -2 + 2  cr r  So the relations between Einstein tensor and are gained:

v2 G 3 00 r2 c 2

a v2  Gab  22 + 2 2  g ab （ a,b=1,2,3） rc r c 

Just demand a v2  v 2   - 22 + 2 2  3 2 2 rc r c  r c Einstein tensor is written as below:

1 0 0 0  2   v  0 - 0 0  Gab = 3 2 2  r c  0 0 - 0    0 0 0 -  As we know, energy-momentum tensor conforming to [7] gives: p T（） + v v - p  g abc2 a b ab The nonzero components are listed as:

2 T00   c

2 Tab -   c  g ab （ a,b=1,2,3）

We have accordingly:

1 0 0 0  0 - 0 0  T = c2    ab   0 0 - 0    0 0 0 -  Consequently, matrix components of Einstein field equation are represented as: 1 0 0 0   1 0 0 0  0 - 0 0 2  0 -  0 0  8G2   3 v    4c = 2 2   c0 0 - 0  r c   0 0 -  0      0 0 0 -   0 0 0 -   It directly results in Friedmann equation (  = 0 ):

c2 1 3v 2  8  2 2  Ep x p r c 

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4 Provided that both sides of Friedmann equation are multiplied by V  r3 , we get (6): 3

Mc2 r v 2  = 2  Ep x p 2c  Consider a body in motion relative to coordinate system. Here co-moving coordinates can’t be adopted. So ener- gy-momentum tensor is given by v vxy v z  1  c c c  2 vx v xvx v y v x v z  2 2 2  2 c c c c  Tab = c  2 vy v x v y v y v y v z  2 2 2  c c c c  2 v v vvy v z v  z x z z  c c2 c 2 c 2  One may write down the components (matrix form) of field equation at this point:

v v vxy v z   v x y v z  1   1  c c c   c c c  2v v 2 v v vx v xx y v x v z   v x v x x y v x v z   2   8Gc2 2 c 2 3 v  ccc c 2 c 2 c 2 c2   =    42 2 2  2 cvvvvvvyxyyyz  r c   vvvvvv yxyyyz  2 2 2   2 2 2  c c   ccc c c c  2 2 v v vvy v z v   v v v v y v z v  z x z z   z x z z  c2 2 c 2   ccc c 2 c 2 c 2 

REFERENCES [1] Unruh W.G., Phys.Rev.D, 14 (1976) 870. [2] Verlinde E., “On the origin of gravity and the laws of Newton,” arXiv:hep-th/1001.0785vl preprint, 2010. [3] Jacobson T., “Thermodynamics of : The Einstein ,” arXiv:gr-qc/9504004v2 preprint, 1995. [4] Bekenstein J.D., Phys.Rev.D, 7 (1973)2333. [5] Hawking S.W., Commun.Math.Phys., 43(1975)199. [6] Caldwell R.R. and Stebbins A., Phys.Rev.Lett. 85, (2000)5042. [7] Cahill M.E. and Taub A.H., Commun.Math.Phys., 21 (1971)1~40.

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