- #1
grav-universe
- 460
- 1
Years ago I posted a thread where I solved for the exterior vacuum metric for a static spherical mass using only a single one of the unknown functions, A, B, or D, where D = C r^2, since they are inter-related. A moderator here graciously supplied the EFE's as A, B, and D relate to the energy density and pressures and suggested or asked if the same thing could be done for the interior metric. This sparked an interest in me to attempt it and after a few months I posted a thread with full solutions in the form of crude infinite Taylor series, which were necessary to work around square roots and denominators in the EFE's in order to integrate. The thread was removed by the same moderator stating that the solutions should be in closed form, and suggested referring to the only closed form interior solution known, that for a constant energy density. Okay, fair enough I suppose, so for years I attempted a full solution in closed form to no avail. But along the way, however, I found some particular forms for the energy density distributions that allow full solutions in closed form, so want to post some of those which I have found. I messaged first to make sure everything is up to speed, but was told I must first have an equation of state before I can use those solutions, otherwise it is personal research and verging on personal speculation. I'm not sure what this means. As far as I can tell, the EFE's themselves are the equations of state to begin with. Is that correct? If I assume isotropic pressures and make a coordinate choice and input a form for the energy density distribution directly into the EFE's and find solutions for A, B, and D in closed form, then as far as I know I am just working through the math of the EFE's exactly as they stand according to the mainstream, right?
So in an effort to comply, I would like to know what the equation of state for the EFE's should be, then, or what form it should take, and I will gladly begin my thread with that as well. Is it just the EFE's themselves? I would have started by stating those anyway. Is it the form of the energy density distribution I am solving for? The only form of the energy density distribution known in closed form is that of constant energy density, let's call it J, where J is constant. So let's say I want to add an extra term to that and solve for say, J - K r, where K is also constant. Would this be my equation of state in that case, then, that which I am solving for? I'm not sure if just stating a random energy density distribution to be solved for is the perceived problem here also. I would not not be stating that the energy density distribution must take this form, just what the solutions to the EFE's would be if it did. J and K allow for two parameters in the input instead of just one. If K = 0, then it reduces to just the constant energy density solution, so the known solution is still contained as a subset where K just expands upon that by allowing another possible term. Would starting the thread with the EFE's spelled out for energy density and pressure in terms of A, B, and D, assuming isotropic pressure and stating the coordinate choice, and then stating that we are solving for an energy density distribution of J - K r be enough here to consider the equation of state to be satisfied? What am I missing?
So in an effort to comply, I would like to know what the equation of state for the EFE's should be, then, or what form it should take, and I will gladly begin my thread with that as well. Is it just the EFE's themselves? I would have started by stating those anyway. Is it the form of the energy density distribution I am solving for? The only form of the energy density distribution known in closed form is that of constant energy density, let's call it J, where J is constant. So let's say I want to add an extra term to that and solve for say, J - K r, where K is also constant. Would this be my equation of state in that case, then, that which I am solving for? I'm not sure if just stating a random energy density distribution to be solved for is the perceived problem here also. I would not not be stating that the energy density distribution must take this form, just what the solutions to the EFE's would be if it did. J and K allow for two parameters in the input instead of just one. If K = 0, then it reduces to just the constant energy density solution, so the known solution is still contained as a subset where K just expands upon that by allowing another possible term. Would starting the thread with the EFE's spelled out for energy density and pressure in terms of A, B, and D, assuming isotropic pressure and stating the coordinate choice, and then stating that we are solving for an energy density distribution of J - K r be enough here to consider the equation of state to be satisfied? What am I missing?