Mechanistic Model of Gravitation

Abstract

The theory of elasticity of the defect-free space of events in 4D is proposed as a variant of the mechanistic theory of gravity. A 4D displacement vector is determined, the fourth component of which is the local non-uniform time of the physical process that generates the gravitational field. A consistent mechanistic theory of physical processes in a defect-free space of events generating a gravitational field is constructed. We believe that any displacement field determines some physical process, which is generated by the corresponding fields of external influences. Each such displacement field can be associated with a tensor stress field, which can be interpreted as a gravitational effect on a test body, as well as a conserved energy-momentum tensor. The proposed theory is radically different from Einstein’s gravitational theory, since a non-zero energy-momentum tensor exists for any physical process determined by equilibrium equations written in displacements, and the Ricci or Einstein (Saint-Venant) tensor for solutions in displacements is always equal to zero. We propose to consider General Relativity (GR) as a variant of the 4D partial theory of elasticity of the defective space of events.

Appendices

Appendix I

1.1 PROPERTIES OF LEVI–CIVITA TENSORS IN 4D

Tensor product of Levi-Civita tensors in 4D:

$$e_{ijmn}e_{pqab}={}\delta_{ip}[\delta_{jq}(\delta_{ma}\delta_{nb}-\delta_{mb}\delta_{na})-\delta_{ja}(\delta_{mq}\delta_{nb}-\delta_{mb}\delta_{nq})+\delta_{jb}(\delta_{mq}\delta_{na}-\delta_{ma}\delta_{nq})]$$

$${}-\delta_{iq}[\delta_{jp}(\delta_{ma}\delta_{nb}-\delta_{mb}\delta_{na})-\delta_{ja}(\delta_{mp}\delta_{nb}-\delta_{mb}\delta_{np})+\delta_{jb}(\delta_{mp}\delta_{na}-\delta_{ma}\delta_{np})]$$

$${}+\delta_{ia}[\delta_{jp}(\delta_{mq}\delta_{nb}-\delta_{mb}\delta_{nq})-\delta_{jq}(\delta_{mp}\delta_{nb}-\delta_{mb}\delta_{np})+\delta_{jb}(\delta_{mp}\delta_{nq}-\delta_{mq}\delta_{np})]$$

$${}-\delta_{ib}[\delta_{jq}(\delta_{ma}\delta_{np}-\delta_{mp}\delta_{na})-\delta_{ja}(\delta_{mq}\delta_{np}-\delta_{mp}\delta_{nq})+\delta_{jp}(\delta_{mq}\delta_{na}-\delta_{ma}\delta_{nq})].$$

Convolutions of Levi-Civita tensors in 4D over one pair of indices:

$$e_{ijmb}e_{pqab}={}\delta_{ip}[3\delta_{jq}\delta_{ma}-3\delta_{ja}\delta_{mq}+(\delta_{mq}\delta_{ja}-\delta_{ma}\delta_{jq})]$$

$${}-\delta_{iq}[\delta_{jp}(\delta_{ma}4-\delta_{ma})-\delta_{ja}(\delta_{mp}4-\delta_{mp})+(\delta_{mp}\delta_{ja}-\delta_{ma}\delta_{jp})]$$

$${}+\delta_{ia}[\delta_{jp}(\delta_{mq}4-\delta_{mq})-\delta_{jq}(\delta_{mp}4-\delta_{mp})+(\delta_{mp}\delta_{jq}-\delta_{mq}\delta_{jp})]$$

$${}-[\delta_{jq}(\delta_{ma}\delta_{ip}-\delta_{mp}\delta_{ia})-\delta_{ja}(\delta_{mq}\delta_{ip}-\delta_{mp}\delta_{iq})+\delta_{jp}(\delta_{mq}\delta_{ia}-\delta_{ma}\delta_{iq})]$$

$${}={}2\delta_{ip}(\delta_{jq}\delta_{ma}-\delta_{ja}\delta_{mq})-2\delta_{iq}(\delta_{jp}\delta_{ma}-\delta_{ja}\delta_{mp})+2\delta_{ia}(\delta_{jp}\delta_{mq}-\delta_{jq}\delta_{mp})$$

$${}-[(\delta_{jq}\delta_{ma}-\delta_{ja}\delta_{mq})\delta_{ip}-(\delta_{jp}\delta_{ma}-\delta_{ja}\delta_{mp})\delta_{iq}+(\delta_{jp}\delta_{mq}-\delta_{jq}\delta_{mp})\delta_{ia}]$$

$${}={}\delta_{ip}(\delta_{jq}\delta_{ma}-\delta_{ja}\delta_{mq})-\delta_{iq}(\delta_{jp}\delta_{ma}-\delta_{ja}\delta_{mp})+\delta_{ia}(\delta_{jp}\delta_{mq}-\delta_{jq}\delta_{mp}).$$

Convolutions of Levi-Civita tensors in 4D over two, three and four pairs of indices:

$$e_{ijab}e_{pqab}=\delta_{ip}(\delta_{jq}4-\delta_{jq})-\delta_{iq}(\delta_{jp}4-\delta_{jp})+(\delta_{jp}\delta_{iq}-\delta_{jq},\delta_{ip})$$

$${}=3(\delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp})-(\delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp})=2(\delta_{ip}\delta_{jq}-\delta_{iq}\delta_{jp}),$$

$$e_{iqab}e_{pqab}=2(\delta_{ip}4-\delta_{ip})=6\delta_{ip},$$

$$e_{pqab}e_{pqab}=24.$$

Appendix II

2.1 RIEMANN–CHRISTOFFEL COMPATIBILITY EQUATIONS IN 4D

Using equation (15) and results of Appendix I, we can get Riemann–Christoffel compatibility equations

$$R_{pqmn}=\varepsilon_{ia,bj}e_{ijmn}e_{pqab}=0,$$

$$R_{pqmn}=\varepsilon_{ia,bj}\{\delta_{ip}[\delta_{jq}(\delta_{ma}\delta_{nb}-\delta_{mb}\delta_{na})-\delta_{ja}(\delta_{mq}\delta_{nb}-\delta_{mb}\delta_{nq})+\delta_{jb}(\delta_{mq}\delta_{na}-\delta_{ma}\delta_{nq})]$$

$${}-\delta_{iq}[\delta_{jp}(\delta_{ma}\delta_{nb}-\delta_{mb}\delta_{na})-\delta_{ja}(\delta_{mp}\delta_{nb}-\delta_{mb}\delta_{np})+\delta_{jb}(\delta_{mp}\delta_{na}-\delta_{ma}\delta_{np})]$$

$${}+\delta_{ia}[\delta_{jp}(\delta_{mq}\delta_{nb}-\delta_{mb}\delta_{nq})-\delta_{jq}(\delta_{mp}\delta_{nb}-\delta_{mb}\delta_{np})+\delta_{jb}(\delta_{mp}\delta_{nq}-\delta_{mq}\delta_{np})]$$

$${}-\delta_{ib}[\delta_{jq}(\delta_{ma}\delta_{np}-\delta_{mp}\delta_{na})-\delta_{ja}(\delta_{mq}\delta_{np}-\delta_{mp}\delta_{nq})+\delta_{jp}(\delta_{mq}\delta_{na}-\delta_{ma}\delta_{nq})]\}$$

$${}=\varepsilon_{pa,bj}[\delta_{jq}(\delta_{ma}\delta_{nb}-\delta_{mb}\delta_{na})-\delta_{ja}(\delta_{mq}\delta_{nb}-\delta_{mb}\delta_{nq})+\delta_{jb}(\delta_{mq}\delta_{na}-\delta_{ma}\delta_{nq})]$$

$${}-\varepsilon_{qa,bj}[\delta_{jp}(\delta_{ma}\delta_{nb}-\delta_{mb}\delta_{na})-\delta_{ja}(\delta_{mp}\delta_{nb}-\delta_{mb}\delta_{np})+\delta_{jb}(\delta_{mp}\delta_{na}-\delta_{ma}\delta_{np})]$$

$${}+\varepsilon_{ii,bj}[\delta_{jp}(\delta_{mq}\delta_{nb}-\delta_{mb}\delta_{nq})-\delta_{jq}(\delta_{mp}\delta_{nb}-\delta_{mb}\delta_{np})+\delta_{jb}(\delta_{mp}\delta_{nq}-\delta_{mq}\delta_{np})]$$

$${}-\varepsilon_{ia,ij}[\delta_{jq}(\delta_{ma}\delta_{np}-\delta_{mp}\delta_{na})+\delta_{ja}(\delta_{mq}\delta_{np}-\delta_{mp}\delta_{nq})-\delta_{jp}(\delta_{mq}\delta_{na}-\delta_{ma}\delta_{nq})]$$

$${}=\varepsilon_{pa,bq}(\delta_{ma}\delta_{nb}-\delta_{mb}\delta_{na})-\varepsilon_{pa,ba}(\delta_{mq}\delta_{nb}-\delta_{mb}\delta_{nq})+\Delta\varepsilon_{pa}(\delta_{mq}\delta_{na}-\delta_{ma}\delta_{nq})$$

$${}-\varepsilon_{qa,bp}(\delta_{ma}\delta_{nb}-\delta_{mb}\delta_{na})+\varepsilon_{qa,ba}(\delta_{mp}\delta_{nb}-\delta_{mb}\delta_{np})-\Delta\varepsilon_{qa}(\delta_{mp}\delta_{na}-\delta_{ma}\delta_{np})$$

$${}+\varepsilon_{ii,bp}(\delta_{mq}\delta_{nb}-\delta_{mb}\delta_{nq})-\varepsilon_{ii,bq}(\delta_{mp}\delta_{nb}-\delta_{mb}\delta_{np})+\Delta\varepsilon_{ii}(\delta_{mp}\delta_{nq}-\delta_{mq}\delta_{np})$$

$${}-\varepsilon_{ia,iq}(\delta_{ma}\delta_{np}-\delta_{mp}\delta_{na})+\varepsilon_{ia,ia}(\delta_{mq}\delta_{np}-\delta_{mp}\delta_{nq})-\varepsilon_{ia,ip}(\delta_{mq}\delta_{na}-\delta_{ma}\delta_{nq})$$

$${}=(\varepsilon_{pm,nq}-\varepsilon_{pn,mq})-(\varepsilon_{pa,na}\delta_{mq}-\varepsilon_{pa,ma}\delta_{nq})+(\Delta\varepsilon_{pn}\delta_{mq}-\Delta\varepsilon_{pm}\delta_{nq})$$

$${}-(\varepsilon_{qm,np}-\varepsilon_{qn,mp})+(\varepsilon_{qa,na}\delta_{mp}-\varepsilon_{qa,ma}\delta_{np})-(\Delta\varepsilon_{qn}\delta_{mp}-\Delta\varepsilon_{qm}\delta_{np})$$

$${}+(\varepsilon_{ii,np}\delta_{mq}-\varepsilon_{ii,mp}\delta_{nq})-(\varepsilon_{ii,nq}\delta_{mp}-\varepsilon_{ii,mq}\delta_{np})+(\Delta\varepsilon_{ii}\delta_{mp}\delta_{nq}-\Delta\varepsilon_{ii}\delta_{mq}\delta_{np})$$

$${}-(\varepsilon_{im,iq}\delta_{np}-\varepsilon_{in,iq}\delta_{mp})+(\varepsilon_{ia,ia}\delta_{mq}\delta_{np}-\varepsilon_{ia,ia}\delta_{mp}\delta_{nq})-(\varepsilon_{in,ip}\delta_{mq}-\varepsilon_{im,ip}\delta_{nq})=0.$$

Appendix III

3.1 VARIATIONAL EQUATION OF MECHANISTIC THEORY OF RG

Let us consider the conditional functional (37) and transform it using the procedure of successive integration by parts. We obtain

$$\delta\tilde{L}={}\int\limits_{0}^{ict}\biggl{\{}mc^{2}-\int\limits_{V_{3}}\biggl{[}(C_{ijmn}N_{k}N_{j})R_{m,n}R_{i,k}-\frac{1}{2}C_{abmn}R_{a,b}R_{m,n}\biggr{]}dV_{3}\biggr{\}}\delta\Lambda\,dx_{4}$$

$${}-\int\limits_{0}^{ict}\int\limits_{V_{3}}[C_{ijmn}R_{m,n}\delta R_{i,k}(\delta_{jk}^{\ast}+N_{j}N_{k})+\Lambda(C_{ijpq}N_{k}N_{q})R_{p,k}\delta R_{i,j}$$

$${}+\Lambda(C_{iqmn}N_{j}N_{q})R_{m,n}\delta R_{i,j}-\Lambda C_{ijmn}R_{m,n}\delta R_{i,j}]dV_{3}\}\,dx_{4}$$

$${}=\int\limits_{0}^{ict}\biggl{\{}mc^{2}-\int\limits_{V_{3}}\biggl{[}(C_{ijmn}N_{k}N_{j})R_{m,n}R_{i,k}-\frac{1}{2}C_{abmn}R_{a,b}R_{m,n}\biggr{]}\,dV_{3}\biggr{\}}\delta\Lambda\,dx_{4}$$

$${}-\int\limits_{0}^{ict}\int\limits_{V_{3}}\biggl{[}C_{ijmn}R_{m,n}\delta(R_{i,k}\delta_{jk}^{\ast})+C_{ijmn}R_{m,n}\delta(R_{i,k}N_{j}N_{k})$$

$${}+\Lambda(C_{ijpq}N_{r}N_{q})R_{p,r}\delta(R_{i,k}\delta_{jk}^{\ast})+\Lambda(C_{ijpq}N_{r}N_{q})R_{p,r}\delta(R_{i,k}N_{j}N_{k})$$

$${}+\Lambda(C_{iqmn}N_{j}N_{q})R_{m,n}\delta(R_{i,k}\delta_{jk}^{\ast})+\Lambda(C_{iqmn}N_{j}N_{q})R_{m,n}\delta(R_{i,k}N_{j}N_{k})$$

$${}-\Lambda C_{ijmn}R_{m,n}\delta(R_{i,k}\delta_{jk}^{\ast})-\Lambda C_{ijmn}R_{m,n}\delta(R_{i,k}N_{j}N_{k})\biggr{]}dV_{3}\}\,dx_{4}$$

$${}=\int\limits_{0}^{ict}\biggl{\{}mc^{2}-\int\limits_{V_{3}}\biggl{[}(C_{ijmn}N_{k}N_{j})R_{m,n}R_{i,k}-\frac{1}{2}C_{abmn}R_{a,b}R_{m,n}\biggr{]}dV_{3}\biggr{\}}\delta\Lambda\,dx_{4}$$

$${}-\int\limits_{0}^{ict}\int\limits_{V_{3}}\bigl{[}(C_{ijmn}R_{m,n}+\Lambda C_{ijpq}N_{r}N_{q}R_{p,r}+\Lambda C_{iqmn}N_{j}N_{q}R_{m,n}-\Lambda C_{ijmn}R_{m,n})\delta(R_{i,k}\delta_{jk}^{\ast})$$

$${}+(C_{ijmn}R_{m,n}+\Lambda C_{ijpq}N_{r}N_{q}R_{p,r}+\Lambda C_{iqmn}N_{j}N_{q}R_{m,n}-\Lambda C_{ijmn}R_{m,n})\delta(R_{i,k}N_{j}N_{k})\bigr{]}dV_{3}\biggr{\}}\,dx_{4}$$

$${}={}\int\limits_{0}^{ict}{\int\limits_{V_{3}}}(C_{ijmn}R_{m,nk}+\Lambda C_{ijpq}N_{r}N_{q}R_{p,rk}+\Lambda C_{iqmn}N_{j}N_{q}R_{m,nk}-\Lambda C_{ijmn}R_{m,nk})\delta_{jk}^{\ast}\delta R_{i}\,dV_{3}\,dx_{4}$$

$${}+\int\limits_{0}^{ict}{\int\limits_{V_{3}}}(C_{ijmn}R_{m,nk}+\Lambda C_{ijpq}N_{r}N_{q}R_{p,rk}+\Lambda C_{iqmn}N_{j}N_{q}R_{m,nk}-\Lambda C_{ijmn}R_{m,nk})N_{j}N_{k}\delta R_{i}\,dV_{3}\,dx_{4}$$

$${}-\int\limits_{0}^{ict}\mathop{{\int\!\!\!\!\!\int}\mskip-21.0mu \bigcirc}\bigl{(}C_{ijmn}R_{m,n}+\Lambda C_{ijpq}N_{r}N_{q}R_{p,r}+\Lambda C_{iqmn}N_{j}N_{q}R_{m,n}-\Lambda C_{ijmn}R_{m,n}\bigr{)}n_{j}\delta R_{i}\,dF_{3}\,dx_{4}$$

$${}+\int\limits_{V_{3}}\bigl{(}C_{ijmn}R_{m,n}+\Lambda C_{ijpq}N_{r}N_{q}R_{p,r}+\Lambda C_{iqmn}N_{j}N_{q}R_{m,n}-\Lambda C_{ijmn}R_{m,n}\bigr{)}N_{j}N_{k}\delta R_{i}]dV_{3}|_{0}^{x_{4}=x_{4}}$$

$${}+\int\limits_{0}^{ict}\biggl{\{}mc^{2}-\int\limits_{V_{3}}\biggl{[}(C_{ijmn}N_{k}N_{j})R_{m,n}R_{i,k}-\frac{1}{2}C_{abmn}R_{a,b}R_{m,n}\biggr{]}dV_{3}\biggr{\}}\delta\Lambda\,\,dx_{4}=0.$$