In a four dimensional world any point has four coordinates (x,y,z,t). Then two different points will be denoted by (x1,y1,z1,t1) and (x2,y2,z2,t2); and the invariant quantity is ds2=dx2+dy2+dx2 - dt2. If two points are very close to each other dx= x1-x2 and similarly for all other coordinates.

According to the theory of relativity, two different observers who are related by a Lorentz transformation will observe same value of ds2 but different values for dx,dy,dz and dt. That means any numerical computational code should make sure that ds2 remains an invariant quantity while computing physical phenomenon. Otherwise, if you compute in the rest frame of a particle, you will surely get a different clock reading in comparison with that of an observer who is in motion relative to the particle, and also get different values of ds2 which is not acceptable.

Your answer is very interesting;

Many thanks, dear Biswajoy Brahmachari.

What physical phoenomenon are you interested in? In the CFD (non-relativistic) field we have a computational time step that is usually a user-dependent choice (dictated by numerical constraints) but we have also characteristic times that depend on the physics of the flow problem.

As an example, you can see the Reynolds number as the ratio of characteristic diffusive and inertial times. On the other hand, the CFL number is the ratio of the computational time step and the permanence time of a particle in a cell.

Could you should further elaborate your question?

Many thanks, dear Prof. Filippo Maria Denaro for your answer.

For your question: I am interested in the modeling of completely devloped flow in real geometries.

Please Professors, in maintaining ds^2 invariant,will it result to keeping code time and real particle time almost equal? does particle filters help with the computational cost

For example, Galilean invariance property is still a debated issue in CFD codes, see https://home.aero.polimi.it/quadrio/papers/2013-JCP.pdf

Thank you Professor Filippo (Sir).,i found the work enlightening..

Neither term makes sense. There's only proper time, that's invariant under global Lorentz transformations, that map an inertial frame to another and transforms in a well-defined way under local Lorentz transformations.