Jorma Rantakivi

Yleinen suhteellisuusteoria antaa taivaankappaleen radan Eulerin yhtälöiden ratkaisuna. Ratkaisu on sellainen ajasta riippuva paikkavektori, joka minimoi radanpituuden kaareutuvassa avaruudessa. Samoja Eulerin yhtälöitä voidaan katsoa myös toisesta näkökulmasta. Ne voidaan tulkita yhtälöryhmäksi kappaleen kiihtyvyyden koordinaattiakselien suuntaisille komponenteille kappaleen nopeuden ja paikan funktiona suoraviivaisessa Eukleideen avaruudessa. Kiihtyvyydet voidaan ratkaista yhtälöryhmästä numeerisesti. Koska kiihtyvyys on sama kuin voima massayksikköä kohti, saadaan tulokseksi kappaleeseen vaikuttava relativistinen voima. Tällä tavalla voidaan laskea useiden kappaleiden radat yhtäaikaisesti yleisen suhteellisuusteorian mukaisesti. Kirjassa on esitetty tarkempi laskenta-algoritmi, FORTRAN-kielinen ohjelma ja useita laskettuja esimerkkitapauksia.

General relativity gives the orbit of a celestial body as a solution to Euler's equations. The solution is a time-dependent position vector that minimizes the orbital length in curved space. The same Euler equations can also be viewed from another perspective. They can be interpreted as a group of equations for components parallel to the coordinate axes of the acceleration of the body as a function of the velocity and position of the body in a rectilinear Euclidean space. Accelerations can be solved numerically from the group of equations. Since the acceleration is the same as the force per unit mass, the result is the relativistic force acting on the object. In this way, the orbits of several bodies can be calculated simultaneously according to general relativity. The book presents a more detailed calculation algorithm, a program in the FORTRAN language and several calculated example cases.

General relativity gives the orbit of a celestial body as a solution to Euler's equations. The solution is a time-dependent position vector that minimizes the orbital length in curved space. The same Euler equations can also be viewed from another perspective. They can be interpreted as a group of equations for components parallel to the coordinate axes of the acceleration of the body as a function of the velocity and position of the body in a rectilinear Euclidean space. Accelerations can be solved numerically from the group of equations. Since the acceleration is the same as the force per unit mass, the result is the relativistic force acting on the object. In this way, the orbits of several bodies can be calculated simultaneously according to general relativity. The book presents a more detailed calculation algorithm, a program in the FORTRAN language and several calculated example cases.