In Newtonian mechanics, the dynamic equation F=ma is Galileo invariant. Its quantization equation, Schrodinger equation, is also Galileo invariant.
In Newtonian gravity theory F=Gm_1m_2/r^2, or its field equation is Poisson equation that is also Galileo invariant. The geometrization equation of Poisson equation should keep Galileo invariance.
Both quantization from the Newtonian dynamic equation and geometrization from the Newtonian gravity field equation should be regarded as generalizations from so-called "classical" theories. It seems to keep their original invariances, namely, Galileo invariance.
Why is Einstein field equation Lorentz invariant?
From this point of view, the geometrization field equation of Newtonian gravity theory should has Galileo invariace instead of Lorentz invariance.
I agree with Niz Nl, the main reason that i see for Lorentz invariance is that this is truly the symmetry of nature (Galileo's symmetry is just an aproximation). Therefore, field equations must respect this relativistic symmetry and that's why you should add by hand the requirment of Lorentz invariance.
I would like to add a little detail here. Recall that Einstein field equations have wider symmetry, they are "generally covariant" which means that they are invariant under general coordinate transformations of the space time (technically called difeomorphism of the space time manifold). I dont have a clear answer for whether this transformations should preserve the metric tensor or not (in order to reduce the symmetry group to the so called isometry group of the space time manifold). The reason for this is simple, it is not clear what is the metric if you dont know the solutions to the equations (it is not even clear if the equations have solutions, in a general manifold). From this point of view you can recover "lorentz invariance" just locally.
Greeting
Raúl
I agree with Niz Nl, the main reason that i see for Lorentz invariance is that this is truly the symmetry of nature (Galileo's symmetry is just an aproximation). Therefore, field equations must respect this relativistic symmetry and that's why you should add by hand the requirment of Lorentz invariance.
I would like to add a little detail here. Recall that Einstein field equations have wider symmetry, they are "generally covariant" which means that they are invariant under general coordinate transformations of the space time (technically called difeomorphism of the space time manifold). I dont have a clear answer for whether this transformations should preserve the metric tensor or not (in order to reduce the symmetry group to the so called isometry group of the space time manifold). The reason for this is simple, it is not clear what is the metric if you dont know the solutions to the equations (it is not even clear if the equations have solutions, in a general manifold). From this point of view you can recover "lorentz invariance" just locally.
Greeting
Raúl
My understanding of Einstein's field equation, is that by 'definition' Einstein choose mathematical equations such that this would be true. It's the basic concept behind Special Relative, where General Relativity is sort of an extension of Special Relativity. See http://en.wikipedia.org/wiki/Lorentz_covariance, which does a 'poor' job for invariant, as it focuses on covariance. Your question might then be reworded?
Do read a translation of Einstein's original German Paper on General Relativity. It's an amazing easy read, for experts or novices.
Thank you for your answer. However, General relativity (GR) should be an extension of Newtonian gravity theory (NGT) instead of Special relativity (SR) . The Lorentz invariance of SR comes from the Maxwell's equation, but the equation of NGT is Poisson's equation which is Galileo's invariance. So, GR seems to be Galileo's invariance locally in the limit of weak field instead of Lorentz invariance.
Dear Shi-Dong Liang
I think i have a quite different opinion about your last statement. First i like to add a single word in what respects to the relation between electrodynamics and special relativity. Lorentz invarianze HISTORICALLY comes from Maxwell's formulation of electrodynamics. BUT it is not a logical consequence of the latter, in fact you can forget everything and just impose Lorentz invariance as a "mathematical principle" that defines kinematics and then derive electrodynamics (and the rest of physics as well) from that assumption (of course complemented with the corresponding action principle).
The last paragraph was just a minimal difference, i think. What i find more interesting for this discussion is what has to do with general relativity as a limit of Galileo's symmetry or Lorentz symmetry. As i said in my previous post i dont have this issue clear. Explicity i said " I dont have a clear answer for whether this transformations should preserve the metric tensor or not (in order to reduce the symmetry group to the so called isometry group of the space time manifold)... ". "From this point of view you can recover "lorentz invariance" just locally".
I have just one argument in favor to special relativity as the correct limit to the general theory instead to "Galileo". In some situations (QFT situations most of the time) it's convenient to asume the space-time as a general "curved" manifold (technically i'm thinking in a topologically non trivial space-time). So acording to this assumption you should also demand causality conditions on the fields in order to get a consistent theory. And it is well known that the correct causality conditions come from special relativity via Lorentz invariance, not from "Galileo". Think just as an example in the relation between causality and antiparticles.
Well i have nothing more to say on this, hope to read some answers to this.
Greetings
Raíul
In Newton's gravitation, space and time are separate quantities, while in Einstein's general relativity, space-time is a single quantity-the coordinate of the system with four components. In Einstein's theory, objects are in the Minkowski space and the invariance of distance in this space gives the Lorentz invariance. If the speed of the object is very low, then the Galileo invariance is a good approximation to the Lorentz invariance. It is just a matter of which space you are placing your objects in and whether you consider the full relativistic effect or not.
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