a metric describing gravitational field energy

*[email protected] 1 A metric describing gravitational field energy Wolfgang Salm* We consider spherically symmetrical stationary, spatially extended mass distributions such as elliptical galaxies and, as a limit case, spatially limited masses such as the sun in the planetary system and ask for the positions, at which the units of length and time are fixed; in our approach, we investigate a metric of the spacetime where all these positions can be chosen equivalently. Based on this metric, we compute the equations of geodetic lines and Einstein's field equations. For spatially extended mass distributions, the field equations for weak fields yield the mass density profile of the Singular Isothermal Sphere (SIS) and a stress-energy tensor containing gravitational field energy. If the units of length and time are calibrated at infinity, the time scale of the outer Schwarzschild metric is obtained. We examine non-relativistic circular motions, light deflection and the Shapiro time delay in radially symmetric, weak static gravitational fields. For the solar system and for pulsar systems, we reproduce the predictions of general relativity which are based on the Schwarzschild metric. In systems with spatially extended mass density profiles, however, the deflection of light is increased, and the velocities of circularly orbiting stars in galaxies are approximately independent of their radius. Some effects such as those observed in colliding galaxy clusters are qualitatively interpreted without presupposing the existence of dark matter. 1. Introduction and preview The Schwarzschild metric has a singular significance: Considering strong fields, it describes an event horizon and essential properties of black holes. For weak fields, it transitions into the metric – independent parameterized post-Newtonian approximation (PPN). Here, the parameter of the spatial curvature generated per unit mass, , is strongly confirmed by observations in planetary and pulsar systems. On the other hand, the Schwarzschild metric has some shortcomings: First, describing the structure of the spherically symmetric empty spacetime in the environnement of a stationary central mass it does not include the energy density of the gravitational field. Substantial problems arise when gravitational field energy has to be incorporated in the Einstein field equations; as an example, we mention Misner, Thorne and Wheeler , that gravitational field energy cannot be localized because it locally vanishes in a co-moving frame of reference. This is true in general; but here, we restrict ourselves to spherical coordinate systems, in which the central mass distribution is fixed. As noted in Box 23.1 of Misner et al., under these conditions, gravitational field energy of the system is well defined and not vanishing. It is difficult to argue that, in the Schwarzschild metric, the approximation of gravitational acceleration for weak fields and thus the gravitational field aligns with classical mechanics, but that the square of this acceleration, which in Newtonian mechanics is a measure of the energy density of the gravitational field, must be zero. Second, the radial coordinate velocity of light depends on the potential of the surrounding gravitational field ; therefore, natural constants associated with the coordinate velocity of light, expressed in SI units, are also dependent on the gravitational potential. In the relativistic representation of electrodynamics, the speed of light is set. However, if holds in the gravitational field at the location of an observer, then according to the Schwarzschild metric, deviates from 1 for . Third, the definition of the unit of time, which refers to a position in infinity, does not correspond to the type of time measurement as it is actually made, see section 2. In our approach, we first ask about the position in the gravitational field inside and outside mass distributions, where the units of time and length are fixed; we assume that all these reference positions are equivalent. So, our metric obeys a gauge transformation when this position in the gravitational field is changed. This distinguishes it from the inner and outer Schwarzschild metric. Second, we wish that the radial coordinate velocity of light in the stationary gravitational field has the constant value , regardless of the gravitational potential. Both demands are in line with a principle of relativity, which states that there is no preferred position in the gravitational field where the units of length and time are fixed. In section 2, we propose a suitable metric in spherical coordinates. In section 4, using our metric, we present the field equations in order to get insight in the stress-energy tensor. We consider the quadratic terms and show, that they include the energy and pressure of the gravitational field. 2 In section 5, based on our metric, we study the nonrelativistic circular motion of bodies in the solar system and in elliptical galaxies. In section 6 and 7, we examine the deflection and time delay of light in spherically symmetric stationary gravitational fields. If we consider spatially confined central masses such as the sun in the planetary system, our results coincide with those obtained using the Schwarzschild metric and the PPN approximation; if we use our metric in the case of extended mass distributions, such as galaxies, circular velocities of rotating stars become approximately independent of the radius and the deflection of light is magnified, even outside of mass energy distributions. 2. Metric in spherical coordinates A general approach for an isotropic and time-independent metric in spherical coordinates is given by the standard form for metrics with line element ds : (1) and are the scale factors of length and time, c is the constant vacuum velocity of light. This metric is appropriate to describe the solar system or spherical galaxies as stationary systems. In section 8 we are interested in effects that are typically attributed to Dark Matter; we therefore assume here . First, we consider the radial dependence of . Based on the definition of the unit of time in the SI system, an observer localized at any position in the gravitational field with distance from the origin notes, that one second has passed if he has counted 9192631770 oscillations at his local Cs-133 atomic clock. In accordance with this definition of the unit of time, we wish to dispose over the position where . For spatially extended mass distributions such as within galaxies, we use (2) with K > 0 and with arbitrary reference position . Eq. (2) meets the following condition: If the reference position is changed from to and then from to , the corresponding potential differences in the gravitational field add up, see section 4. in (2) has to depend on the mass of the system and of its distribution in space. With we denote the part of a mass distribution that lies within the radius . We are looking for a formula that combines the mass with the number . Dimensional analysis provides as solution . (3) Here, is the constant of gravitation. If the mass with finite radius is localized in the origin, so that , then is constant. Neglecting the masses of the planets, this situation applies in the solar system. Outside of mass distributions we write (4) with constant . Considering light propagating in radial direction ( ), (1) becomes . (5) Here, the distance traveled per time that an observer in a gravitational field attributes to light depends on and herewith on the potential of the gravitational field . Expressed in SI units of time and length, electrodynamic terms such as the vacuum permittivity and the finestructure constant , which are coupled to the coordinate speed of light, also depend on the gravitational potential. Following our principle of relativity, we wish, that the radial coordinate velocity of light does not depend on the position in the gravitational field. So, we set . (6) The condition (6) differs from the outer Schwarzschild metric where applies, describing empty space. To compare both formulas, in a illustrative view, let's look at two corresponding geographical projections of the curved surface of the Earth. To do so, we write the line element that describes the projected length scales on the two rolled out maps, . Then, (6) corresponds to the conformal Mercator map, while the Schwarzschild condition corresponds to the distorting but area-preserving projection of Archimedes. We support (6) with the following reasoning: For distances approaching infinity, the Schwarzschild metric is asymptotically minkowskian and a distant observer is inertial. Within the framework of special relativity, each inertial observer assigns time and length scales modified by the same factor to the system of a body that moves at the speed compared to the local system of the observer. This applies to all speeds of the body. It remains valid when considering differencies of velocities. So, a distant inertial observer assigns equal changes of the time and length scales to the system of a body falling freely in a gravitational field. According to the weak principle of equivalence, regions, where masses are accelerated, inherently equal gravitational fields; so, as seen by an distant inertial observer, the scales of time and length of the body at fixed position in the gravitational field should be modified by the same factor , in deviation from the Schwarzschild metric. if , and also for weak gravitational fields, if we approximate (2): . (7) If is chosen in infinity, (7) turns into (8) and equals the time scale factor of the Schwarzschild metric. If , we obtain a horizon , just like with the Schwarzschild metric. There, the radial coordinate speed (5) of a light ray near approaches zero. But here (5) remains 3 constant, while both the spatial and time scales increase, so that when viewed from a distance , the light also no longer seems to 'move ahead'. 3. Radial accelation with condition In order to compute the radial acceleration of a body rotating circularly in a spherically symmetrical gravitational field we write the equations of geodesic lines with Christoffel symbols ( : . (9) We integrate the first equation ( , ; (10) with an appropriate constant of integration, we get . (11) So, we connect with the proper time of the moving body, . Using with , the radial equation ( of (9) is . (12) Using (11), we get (13) with the angular velocity . Presupposing location independent timeflow , the angular velocity as ascribed to a body moving at a location with with higher gravitational potential thus appears to be increased. The radial acceleration without the centrifugal term in (13) is . (14) First, outside of the constant mass , assuming spherical symmetry, we insert the metric , (8), in (14): . (15) In the solar system, since , the deviations with factor from the Newtonian law will be hardly detectable, since actual tests of general relativity in the solar system have an accuracy of only . So, for a mass which is located in a small area near the origin, we get approximatively the classical Newtonian potential with . (16) Now, we use the metric , (2). For spherically symmetric radial mass distributions, we have the density profile as calculated in (25); herewith, we get the acceleration (17) with constant . This result is consistent with the classically calculated acceleration and confirms the metric (2). Outside of the mass distribution, there is and (17) merges into the Newtonian acceleration . Based on (17), the potential is . (18) For , the potential is approximated by (19) merging into the potential for spatially limited masses. 4. Field equations with condition Unlike the condition of the outer Schwarzschild metric, describing empty spacetime, due to (6), in our approach, spacetime has to contain energy also outside of masses. In the following, we examine whether gravitational field energy may contribute to this. We note the Einstein field equations with Ricci tensor , . (20) Using (6) and assuming spherical symmetry, the stress-energy tensor on the right side is . (21) Here, is the energy density, is the radial pressure component and are the lateral pressure components perpendicular to . From the very beginning, we don't specify here what kind of energy and pressure are involved in the field equations; so, we do not presuppose isotropic pressure , as it is e.g. present in ideal liquids. We calculate the elements of the Ricci tensor using (6), see Appendix A: (22) (23) . (24) The classical formulas for describing gravitational field energy assume that the unit of time is independent of the gravitational potential. In the limiting case , the classical formulas should also emerge here. To check that, in the following, we will limit ourselves to considering solutions of the field equations in the vicinity of . In (24), with , . So, . Herewith, and in (20) is vanishing. We compare the first terms in (22) and (23) with (14) and conclude that they express the radial change rate of the acceleration and thus the inhomogeneity of the gravitational field. Restricting us to small environments , we may neglect the inhomogeneity of the field and consider only the second terms in (22) and (23). Then, . This is 4 confirmed by the calculation in Appendix A. Eqs. (22) and (23) are solved by the Singular Isothermal Sphere (SIS) mass density profile (25) with and the condition (26) see Appendix A. So, (27) Is constant. The radial acceleration in (14) and the Ricci tensor elements in (22) and (23), which are directly connected with measurable quantities, contain only derivatives of ; since is constant, these quantities are invariant if we add to a term expressing the change in reference position from to arbitrary . This gauge transformation ensures that in (18) the potential differences are additive and confirms the distance law in (2). To interprete (26), we note that with a given SIS density profile (26), an adiabatic change in mass density can only be achieved by changing and herewith the total mass . So, the pressure work required for enlarging the density is equal to the energy required to enlarge the mass . Multiplying the field equation (28) with and using (17), we get . (29) In the post-Newtonian approximation, developing in a power series, the quadratic term of the Ricci tensor is assumed to describe gravitational field energy density. This is consistent with our approach: the left side of (29) consists entirely of such a quadratic term; but it is not pushed to the right side. Using (17), (25) and (29), we calculate the gravitational pressure which is produced by outside : (30) To interprete this result, we break down the mass distribution into spherical shells with thickness and examine them independently of each other: Inside of each bubble, the gravitational field energy is vanishing, outside it is negative. If we enlarge the bubble adiabatically by without changing the mass volume and mass energy of the bubble itself, the field outside of does not change; this follows because (22) represents a form of the radial part of the Laplace operator with (18), so that , (17), depends only on the mass inside in (3). By increasing the energy density within the spherical mass shell with thickness from negative values to zero, the increase of the gravitational energy of the system, , is completely provided by the pressure work . Using (19), (29) is . (31) Analogous to the procedure in classical theory, we consider the integrated gravitational energy of a radially symmetric mass with spatially limited distribution . For a reference position outside the mass distribution, we get . (32) The field equation for changes to the relation between the two formulas used in classical theory to express gravitational field energy . There, both equations are connected via the Poisson equation which thus appears as classical limit of the field equation (28). Let's take a look at the cosmological constant in our approach. Lovelock's theorem states that any field equation in four dimensions with the energy-stress tensor should be able to be written in the form (33) with constant . This assumes a metric , whose units are calibrated at a fixed point in spacetime. In our approach, we we can split according to with , ) (34) with and , ,0,0). (35) So, = . For , in linear approximation, ; there is no dark energy in our approach. 5. Nonrelativistic circular motions We consider a spherically symmetric system of slowly circularily rotating masses with radial mass density profile . In the framework of classical theory, using the same time scale with variable for all bodies and thereby assuming the centripetal acceleration , one can assign a density profile to each velocity profile of the rotating mass elements according to . (36) Here, the density profile in (25), , is consistent with radius independent velocities and with constant in (3). Only if, as in the solar system, there is a pointlike mass in the center of an otherwise empty space, using the timescale factor as in the Schwarzschild metric, the velocities of planets vary according to . In our approach, starting from in (2) for weak fields, we get the density profile (25) as a solution of (28) and thus a constant value as in (36). But here, in (13), the 5 centrifugal term referes to the proper time of the bodies of the rotating system. An observer, measuring velocities in his local reference system, neglecting the dependence of the time unit of the gravitational potential, measures an increasing velocity . (37) For Saturn S, rotating at half distance to the edge of the solar system with distance and the circular velocity , we get and . In the outer solar system, we expect no measurable deviations from classical gravitation, since in (15), the second-order approximation of the radial acceleration makes an immeasurably small contribution. In order to estimate in a spherical galaxy, presupposing isothermal mass profile, we consider a star rotating circularly with a typical velocity and with radius m around the galactic centre. Using (36), we get . With (2), the time scale of a star relative to is what corresponds to the increased orbital speed assigned to star . With increasing velocities of bodies, we expect increased values of und when we look at galaxy clusters with high velocities of the galaxies in the cluster. 6. Light deflection in a radially symmetric gravitational field In the PPA, the deflection of light and its time delay in gravitational fields are discussed, introducing the parameter , using and , thereby presupposing that the PPA parameter . A value is compatible with the approximation of the Schwarzschild metric for weak fieds. Indeed, no deviations from that condition have been observed inside and outside of the solar system. Can we introduce an alternative metric that, under appropriate conditions, gives the same predictions as the successful Schwarzschild metric? To discuss this question, in a thought experiment we consider two observers who are ´below´ and ´above´ in the radially symmetrical gravitational field outside of a mass distribution. Assuming the Schwarzschild metric, the observer ´above´ attributes a stretched unit of time and a shortened unit of length to the observer ´below´ and adjusts the coordinate speed of light ´below´ to these local units. On the other hand, based on the metric (6), as seen from ´above´, light propagates ´below´ at the same rate as ´above´ since both the distance to be covered and the time available appear to be increased by the same factor. ´Below´, due to (2), the scaling factor is reduced, corresponding to the enlarged units of time and length; therefore, the covered distance per time interval of light traveling ´below´, as measured ´above´ decreases in two ways: . (38) Both approaches coincide describing the deflection of light in weak gravitational fields outside of mass distributions, but with different assumptions regarding the coordinate speed of light. We will now quantitatively examine this illustrative view. Restricting ourselves to an approximately radial path of the light with vanishing orbital angular momentum and with , the radial equation ( of (9) based on (1) is: . (39) With (6), using with constant c, we have . (40) Now, we use ( 11) to get . (41) With (2), we calculate the radial acceleration in the radially symmetric field of a mass distribution according to (25): . (42) Analogous to (15), outside of the spatially limited mass , (42) converges towards . (43) Based on (6), we calculate the deflection angle of light passing a mass along an approximately straight line at a minimal distance according to . (44) Here, is the component of acceleration , which is perpendicular to the light path and acts at a distance from the center of the mass. In accordance with typical experiments testing general relativity in the solar system with central mass , using (43), we get for the angle of the light deflection . (45) Now, we consider weak lensing by extended mass distributions such as galaxies or galaxy clusters. If the light deflection of the radially symmetrical mass distribution is calculated as linearly composed of the deflection due to the individual mass elements as in the thin screen approximation, then in (45) only is to be replaced by . Here, however, we insert the acceleration of (17) in (44), which is caused by the entire mass at each point of the light path. So, we calculate the light deflection (see Appendix B) . (46) The same mass on the one hand concentrated in the origin and on the other hand radially symmetrically distributed with density profile leads to an increased deflection angle of the light. Considering Einstein rings of background objects which are produced by radially symmetric lensing galaxies, one can simultaneously measure the angle and the rotation 6 velocity (36) in the lens via the stellar velocity dispersion. In our approach the following relationship arises: (47) 7. Shapiro time delay If one assumes within the framework of the Schwarzschild metric that the coordinate speed of light is independent of the gravitational potential, then the time delay when crossing a weak radially symmetric gravitational field is zero (see Appendix C). Here, using (6), we calculate the time delay of light in a radially symmetric gravitational field of a spatially limited mass : We start with (41) and we assume, that the light passes approximately along a straight line , starting in distance from the central mass . For the time delay, as seen from a position far away from , we get (see Appendix D) . (48) Hence, considering signals emitted from the earth and reflected within the solar system, we reproduce the Shapiro formula in accordance with typical experiments that test general relativity. Outside the solar system, a Shapiro time delay of has been observed by Kramer et al. in the radiation of the double pulsar PSR J0737-3039 . Testing the theory of general relativity in this regime of locally small, but strong gravitational fields, they found excellent agreement of the unmodified theory with their observations. We also expect this result in our approach, since neutron stars have very compact, spatially limited masses, so that (43) applies. Improving measurements of the Shapiro time delay may make it possible to test the second-order approximation of the radial acceleration in (15). 8. Some notes on dark matter As a strongly simplified model of a spherically symmetrical galaxy, we consider a gravitationally bound, undisturbed accumulation of matter in thermal equilibrium with temperature . This idealised system results in a radial density profile of the mass distribution according to . In our approach, the same density distribution is obtained in (25) as a solution of Einstein's field equations, taking into account our assumptions about the underlying metric. In these approaches, dark matter is not needed to explain approximately radial-independent rotation speeds of circularly rotating particles with different radii. However, if one assumes the Schwarzschild metric as the solution of the field equations with vanishing stress-energy tensor outside a massive galactic bulge in a near-vacuum, then radiusindependent circular velocities of stars in galaxies and the magnified weak galaxy lensing effect usually are justified by the existence of dark matter surrounding galaxies as spatial halos. A number of proposals, which are based on the set of the equations of general relativity, have been made to explain observed flat rotation velocities without claiming the existence of dark matter. In the Milky Way as a flat spiral galaxy Crosta et al. investigate the circular motions of the stars without requiring dark matter. A model of flat spiral galaxy rotation was also proposed by Cooperstock , showing that the rotation curves for the Milky Way and some other spiral galaxies are consistent with the mass density distributions of the visible matter. Our elementary approach, which is limited to approximately spherically symmetric systems such as elliptical galaxies, allowed us to study not only the motions of bodies in gravitational fields, but also the deflection and time delay of light with hypothetical conclusions regarding dark matter. We would like to address this here. The radial mass density profile of spherically symmetric galaxies, as determined by weak lensing, depends on the deflection angle of the light that originates from the background galaxies and is deflected by the galaxy in question. With an increased deflection angle, it follows, that the radial mass density profile is correspondingly flatter. In line with this, Kyu-Hyun Chae et al. achieved a good fit of the observed mass density profiles of sperically symmetric elliptical galaxies using a two-stage approach based on a profile that adapted to the radial distribution of luminosity in the interior and to the weak lensing in the outer region. While in the interior the density profile , expressed by slightly exceeded the isothermal decline – in nice agreement with our approach - , a reduced value emerged for the outer area, which was attributed to the NFW profile of dark matter. This stage may well disappear if, as in the isothermal profile, the increased light deflection is taken into account by a factor of in (45) and (46). Radial velocity profiles in isolated galaxies, which have been derived by weak lensing data, based on the azimuthally averaged shear of background galaxies, are shown to remain flat for hundreds of kpc , even far outside of observable mass distributions. This is expected in our approach: the background galaxies appear in a magnified angular distance from the galactic centres in comparison to the deflection based on the Schwarzschild attraction. The deflection in (46) is even independent of , if one assumes the SIS profile. In colliding galaxy clusters, maps of the hot colliding gas, of star concentrations and of the effects of weak gravitational lensing of sheared background galaxies behind the clusters can be compared. For example, in the well known Bullet Cluster 1E 0657 – 558, after the collision, the hot gas of the two colliding components is lagging behind the regions of enhanced effects of weak gravitational lensing . Without requiring the existence of dark matter, this is expected in our approach, as according to eqs. (45) and (46) the light deflection due to the weak lensing effect is increased compared to that observed in the solar system. 7 Appendix A: The field equations We start with the Ricci tensor given in : , , ( ). We use (6) and the formulas to calculate the representation (22) – (24) of the Ricci tensor. In (22), and (23) we use (2) with , where . On both sides of the two field equations, we have terms and respectively. We approximate these terms in a corresponding manner by setting . So, for , the derivatives of are and . For we approximate (2) to get . So, we have and . Herewith, the field equation is For , this equation is solved by the mass density profile . The field equation is . For , it is solved by the condition . So, we get . Outside the mass distribution, using (4) with constant , we approximate (2) linearly for small deviations : ; for we get and . As in (29), multiplying the field equation with we get . Here, integrating over the whole space on the left side, but only over the mass-filled region of space on the right side where , we get (32). Appendix B: The light deflection in (46) The middle of the radially symmetrical mass distribution lies in the origin . is the point of closest approach of the approximately rectilinear light path to , is the impact parameter, is the distance of a point on the light path to the origin and . The angle between and is . Then, and . The amount of the component of the acceleration perpendicular to the light path, using (41) is . Using we get . Appendix C: Acceleration presupposing constant in the Schwarzschild metric We start with (36), . Using the parameter with constant we get . With the Schwarzschild condition we have . Presupposing weak fields we set : . For small we have . 8 Appendix D: The Shapiro time delay in (48) Here, we use the same designations as in Appendix B and calculate the time delay of light emitted in distance from a central mass and detected at distance . As in sec. 6, we assume that the light propagates along the straight line . We start with (38) using for : , where . Since is constant, we get or . Now, we assume that the light passes near the central, spatially limited mass , so that and ; thus, we have with (8) . For the time delay due to the gravitational field as seen from a position far away from we get by integration . Similar to appendix B ( and ) we calculate . Since we get . If we regard the time delay of radar pulses emitted from the earth at distance , passing near the sun in distance and reflected by a mirror at superior conjunction at distance from the sun back to the earth, we have also to consider the time of flight for the distance ; so, calculating the total time delay for emission and reflexion of the radiation, we arrive at the formula of Shapiro . In our approach, is obtained using the first order approximation of . Developing (38) to second order we calculate the corresponding contribution to the time delay to . A rough estimate with and km yields a correction which is of the first-order time delay. Statements and Declarations No new data were gererated or analysed in support of this research. The author did not receive support from any organization for the submitted work. The author has no relevant financial or non-financial interests to disclose. References Misner, C. W., Thorne, K. S., Wheeler, J. A.: Gravitation (Freemann, San Francisco) (1973), §20.4 Massimo Meneghetti, M.: Introduction to Gravitational Lensing, Lecture notes. Fließbach, T.: Allgemeine Relativitätstheorie, Springer, 6. Auflage (2012) Collet, Th. E. et al: science 360, 1342-1346 (2018) Lovelock,D.: J.Math. Phy. 13, 874 (1972) Kramer, M. et al.: science 314, 97-102 (2006) Crosta, Mariateresa; Giammaria, Marco; Lattanzi, Mario G.; Poggio, Eloisa (August 2020).: MNRAS 496, 2107– 2122. arXiv:1810.04445. Cooperstock, F. I.; Tieu, S. (2007-05-20). International Journal of Modern Physics A. 22 (13): 2293–2325. arXiv:astroph/0610370. Kyu-Hyun Chae et al.: MNRAS 437, 3670-3687 (2014) Mistele, T.: ApJL969 L3 (2024), arXiv:2406.09685v1 Markewitch, M.: Astrophys. J. 604; 596-603, (2004)