In the Tolman-Oppenheimer-Volkoff model of neutron star, the spherically symmetric metrics was written with the help of auxiliary quantity "u". Specifically, "u" was established with the help of g_{rr} component of metric tensor as u = r*(1 + 1/g_{rr})/2 ("r" is the radial distance). Or, the inverse calculation is g_{rr} = -1/(1 - 2*u/r).
In the outer Schwarschild solution for the vacuum, quantity "u" is constant and also component g_{tt} of metric tensor can be given with the help of "u" as g_{tt} = 1 - 2*u/r. This is then used to gauge "u" making the deduction that the gravitational acceleration in general relativity (GR) is given as a = -(c^{2}/2)*(@g_{tt}/@r) = -c^{2}*u/r^{2} in the limit of weak field and this acceleration should be identical to its Newtonian counterpart equal to a = grad(Psi) = @(Psi)/@r = -G*M/r^{2} ("c" is speed of light, symbol "@" is used for the partial derivative, "Psi" is the Newtonian gravitational potential, "G" is the gravitational onstant, and "M" is the mass). Comparing both relations for "a", the numerical value of quantity "u" can be accounted as u = G*M/c^{2}. Or, u = r*Psi/c^{2}.
The GR was initially created as the geometric theory, including the concept of energy and potential, but not mass. However, quantity "u" has often been represented as closely related to mass "M" in the theory of neutron stars. Or, it has been directly identified to mass. (If one chooses such the system of units that G=1 and c=1, then relation u = G*M/c^{2} reduces to the apparent identity u = M.) So, the identification of "u" and "M" has been considered not only in a weak field, but it has been generalized and used in an argumentation everywhere inside the neutron star.
PROPER QUESTION: Is the representation of auxiliary metric quantity "u" as the "mass", or quantity directly related to mass (according to the relation u = G*M/c^{2}) correct? Consequently, must there always be valid u > 0 (since M > 0)? And, why we then demand, in fact, the validity of inequality |g_{rr}| > 1, which is implied by u > 0? What are the arguments, else than already mentioned, for the demand u > 0 and |g_{rr} > 1|?
I note, there was established so-called effective potential in the GR. In the radial problem (with zero angular momentum), this potential can be given as U = c^{2}*sqrt(1 - 2*u/r). We can see that U > 0 even if u < 0. Another question can be: do we need (or why do we need) two concepts of potential in GR? (The effective potential and that corresponding to relation u = r*Psi/c^{2}.)
The statement does not make sense, because the individual components of the metric tensor are not invariant under general coordinate transformations, which are the symmetries of the theory; they're not physical observables. So it doesn't make sense to ask any question about them, but, only, of gauge-invariant quantities that, in general relativity, can only be defined at the boundary of the spacetime or at infinity. The mass, in particular, can be defined at infinity and there it can be shown to be equal to the mass parameter in the Schwarzschild metric.
@Stam Nicolis
Well, it seems that your response is positive in sense that I expected. I am not however sure if I understand it correctly. So, allow me to try to repeat your explanation in other words. I again consider the spherically symmetric, TOV neutron star. The quantity "u" above its surface is constant and can be, in the limit "r --> infinity", gauged with the observable quantity, which is mass of the object. Inside the stellar body, its behavior, as the function of radial distance, can be calculated by integrating the field equations and there is no other constraint (correct?). So, if the equations imply a turn of "u" from positive to negative values in an interval of radial distance inside the star, it makes no sense to reject such the solutions (as astrophysically unacceptable), because "u" cannot be constrained inside the stellar body by an additional demand. (Of course, the metrics resulting from the integration of field equations has to be possible to be taylored with the vacuum metrics above the stellar surface in this surface.)
The statement that a quantity is constant makes sense, only if the quantity is gauge invariant; and, in general relativity, such quantities would then be constants everywhere, including infinity, so it doesn't make sense to discuss them, if they have a non-trivial profile; only their asymptotic values, assuming that it's possible to relate them to conservation laws, have physical meaning. So while g_00 or g_rr is not physical, it can be shown to depend on a quantity that is physical, because it can be shown that this latter quantity is equal to the integral of the 00 component of the energy-momentum tensor at infinity-where spacetime is flat, mass is a Lorentz invariant and that is what establishes the correspondence.
@ Stam Nicolis
I think, I can agree with your explanation except of one detail. O.K., both g_tt = g_00 and g_rr = g_11 can be written with the help of "u" as -1/g_11 = g_00 = 1 - 2*u/r in the region above the stellar surface. And both these functions characterize a certain physical aspect of the real object after their gauging, and this gauging is done, when "u" is gauged - using its correspondence to mass in the limit of flat spacetime in infinity. Your explanation is consistent with the fact that "u" is constant from the outer radius of the object, R_out, to infinity. But "u" is the function of radial distance in the object's interior. (The derivative of "u" with respect of "r" approaches zero in the limit r --> R_out, is zero in r=R_out, and remains zero (therefore "u" constant) in the region r > R_out.)
Anyway, at this stage of discussion, I am interested whether we can or cannot say that "it does not make sense to discuss "u" in the region of stellar interior, because it has a non-trivial profile, there". If the answer is "yes" (it does not make sense), then one cannot use the argument that "u" must be positive, because mass is always positive quantity. If you agree, then our attitude to this problem is the same.
@Aleksei Bykov
At first, my response to the first part of your point (3) follows. Quanity "u" was established (by Oppenheimer and Volkoff in modelling of neutron-star structure) in relation to g_rr component of metric tensor as u=r*(1+1/g_rr)/2. Since g_can vary, also "u" can vary. The reverse relation is g_rr=-1/(1-2*u/r) looks like a form in the Schwarszchild solution and this can perhas cause a confusion. Component g_tt cannot be generally given with the help of "u". It is possible only in the Schwarzschild solution, where g_tt=-1/g_rr, coincidently.
To your poin (2): Yes, to be completely correct, there should have been stated that I used the Schwarzschild coordinates. However, I do not understand why the inequality coud appear only in metric, which is static, spherically symetric ,and VACUUM solution. Namely, there is an example when this is also valid inside the static and spherically symmetric neutron star, i.e. in non-vacuum environment. Yes, the meaning of my question is to find out, if this is accetable (what are the arguments for an invalidation of such a claim). Please, explain this issue in a more detail if possible.
To point (5): I mean so-called effective potential used in the GR to describe the motion (acceleration) of a test particle in a given spacetime. I am afraid that it cannot be regarded only as a conventional quantity. Otherwise, the calculated acceleration would be, in consequece, only "conventional". The second potential is a generalized Newtonian potential,, PSI, when the value of "u" in the Schwarzschild solution is looked for (as the asymptotic value). The second potential then occurs in the formulafor g_tt, specifically g_tt = 1+2*PSI/c^2. I agree that all this is a convent and, hence, PSI is a conventional quantity, but people modelling the neutron star use it to give the argument that u>0 (always), because u=G*M/c^2 and since G>0, c^2>0, and mass M>0, then also u>0. According to these people, we cannot use PSI only if we want, but PSI must always be used in the discussion of the physical representation of"u".
Concerning this as well as the second part of point (3), I comletely agree with you. I asked my question to find a support (another arguments beside I already know) just to be more able to disprove the agrumentation of people modelling the neutron-star structure, who claim that "u" above the stellar surface directly corresonds to the total mass of star and "u" inside the star is represented as the mass within radius "r". So, I welcome your answer which is consistent with my opinion.
@Aleksei Bykov
We know, the components of metric tensor characterize thef geometry of spacetime (or space, in a static case, when space does not change as time is elapsing), whereby the purpose of this is a prediction of observable effects, which are, in the terms of differential calculus, characterized by acceleration (as the second time derivative of spatial coordinates). In the static and spherically symmetric case I consideer, this acceleration (calculated from the appropriate equation of geodesics) equals g_tt/g_rr.
Sorry that I will not accept your suggestion of studying the problem in a dynamic case (when matter moves). I now focuss on how to persuade people that inequailty U < 0 and, this, |g_rr| < 1, because I regard this as crutial point for the astrophysics of static, spherically symmetric objects. I explaineed the sense of all this in other thread https://www.researchgate.net/post/What_is_the_general_definition_if_exists_of_spherical_symmetry_in_a_curved_space_of_general_relativity#view=55f095be6143255eaf8b45c5
@Aleksei Bykov - 2
[Excuse me a wrong manupulation and sending the previous message before it was finished. I finish it now.]
The explanation is in my long message in the discussion below the question I asked in that thread (What is the general definition (if exists) of spherical symmetry in a curved space of general relativity?). Inequality u
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