Editing Gravitational binding energy (section)

==Negative mass component==
{{Cleanup rewrite|there appears to be a serious conceptual inconsistency between the newtonian formula for binding energy and the relativistic concept of Schwarzschild radius.  Perhaps the section would be best deleted.|section|date=August 2024}}

Two bodies, placed at the distance ''R'' from each other and reciprocally not moving, exert a gravitational force on a third body slightly smaller when ''R'' is small. This can be seen as a [[negative mass]] component of the system, equal, for uniformly spherical solutions, to:
<math display="block">M_\mathrm{binding}=-\frac{3GM^2}{5Rc^2}</math>

For example, the fact that Earth is a gravitationally-bound sphere of its current size ''costs'' {{val|2.49421|e=15|ul=kg}} of mass (roughly one fourth the mass of [[Phobos (moon)|Phobos]] – see above for [[Mass–energy equivalence|the same value]] in [[Joule]]s), and if its atoms were sparse over an arbitrarily large volume the Earth would weigh its current mass plus {{val|2.49421|e=15|u=kg}} kilograms (and its gravitational pull over a third body would be accordingly stronger).

It can be easily demonstrated that this negative component can never exceed the positive component of a system. A negative binding energy greater than the mass of the system itself would indeed require that the radius of the system be smaller than:
<math display="block">R\leq\frac{3GM}{5c^2}</math>
which is smaller than <math display="inline">\frac{3}{10}</math> its [[Schwarzschild radius]]:
<math display="block">R\leq\frac{3}{10} r_\mathrm{s}</math>
and therefore never visible to an external observer. However this is only a Newtonian approximation and in [[General Relativity|relativistic]] conditions other factors must be taken into account as well.<ref>{{cite journal | last1 = Katz | first1 = Joseph | last2 = Lynden-Bell | first2 = Donald | last3 = Bičák | first3 = Jiří | date = 27 October 2006 | title = Gravitational energy in stationary spacetimes | journal = [[Classical and Quantum Gravity]] | volume = 23 | issue = 23 | pages = 7111–7128 | doi = 10.1088/0264-9381/23/23/030 | arxiv = gr-qc/0610052 | bibcode = 2006CQGra..23.7111K | s2cid = 1375765 }}</ref>