Don Herbison-Evans ,
donherbisonevans@yahoo.com

revised 8 June 2017, but beware: this work is still in progress

**SUMMARY**

Schrödinger's equation has beeen used to investigate theoretically the bound states in a system consisting of two object matter waves held together by their mutual gravitational attraction, so constituting a Gravitational Atom. As in much of physics, the component objects are anticipated to be complex systems in themselves, but which are small enough and separated by large enough distances for their mutual actions to be described well by their gravitational attraction, and for which other natural forces are only a minor consideration. Examples are

- two Xenon atoms orbiting each other m ~ 10
^{-25}Kg - two grains of sand orbiting each other, m ~ 10
^{-6} - two meteorites orbiting each other, m ~ 10
^{+2} - two asteroids orbiting each other, m ~ 10
^{+14} - a planet orbiting a star, m ~ 10
^{+27} - a star orbiting the central galactic black hole, m ~ 10
^{+31} - a pair of galaxies circling each other, m ~ 10
^{+40} - a pair of galactic clusters circling each other, m ~ 10
^{+50}.

In solving the equations, we find a series of solutions. These are of states corresponding to stationary matter waves that one might infer are stable, and would emit no gravitational radiation except in transitions to other states. Gravitational Atoms in these states may exist and be observable.

**INTRODUCTION**

In the study reported here, we consider theoretically the bound states of a gravitational atom consisting of the matter waves of two objects, one light and one heavy, moving only under the influence of their mutual gravitation. The basic qualitative idea is that the gravitational mass of the rapidly orbiting lighter object is increased as suggested in Special Relativity by its orbital kinetic energy.

We shall use the following variables and constants in MKS units [Allen, 1964] :

M = rest mass of the heavier object, kg

m = rest mass of the lighter object, kg

u = travelling mass of the lighter object, kg

T = kinetic energy of the lighter object, Joules,

V = potential energy, J,

c = velocity of light ≈ 3.0 x 10

G = universal constant of gravitation ≈ 6.7 x 10

h = reduced Planck's constant ≈ 1.1 x 10

**MATHEMATICAL DETAILS**

Quantitatively we may set up the Schrödinger equation to find the characteristic energies of the stationary states of this Gravitational Atom. In this simple exploratory study, we take the approximations that the objects are point masses, that the lighter object moves around the stationary heavier object which is centred on the coordinate origin, and use the simple model of Special Relativity [eg. Schwartz, 2007] so that the mass of the lighter object can be written:

Following the usual development to find the stationary states (e.g [Houston, 1959]) we transform this using de Broglie's relationship:

Thus

B = 2GMm

C = 2mE/h

k = 1,2,3,...

The normalisation condition requires a finite value of

∫ ∫ ∫ ψ.ψ

0 0 0

∫ ∫ ∫ ψ.ψ

-A 0 0

∫ Q

-A

The solutions of the form s

Changing the variable by the substitution

P

P

**GROUND STATES**

From equation (37) a set of Ground States exist when n = 0 (i.e. when k = 2j ):

**GENERAL STATES**

Using equation (38) the general states are found from the solutions of this quadratic equation for v, namely

If instead, we rewrite equation (42) as

**GENERAL STATES FOR LIGHT OBJECTS**

From equation (43) with A.B/n^{2} << 1,
the general states for light objects can be found from

v

E

E

**GENERAL STATES FOR HEAVY OBJECTS**

If, in equation (42) we have n^{2}/(AB) << 1,
and the square root be expanded as a Taylor Series in n^{2}/(AB),
and the higher powers of this quantity truncated, then for small values of n :

= ( (A.B)

≈ ±[ (A.B)

= ± (B/A)

= [ ± h

= ± m.c

**SIZE of LEVEL 1, STATE -1**

For the next level above the ground state :

R

using 4x2

≈ h/(-5.m.E)

Noting that G.M/c^{2} is half the Schwartzchild Radius of the heavier object,
we need

or, assuming that only E is negative :

mM

**THE EXAMPLES**

Thus using equation (55) :

E

E

Using equation (56) :

E_{-1} ≈ - 10^{+47}(m^{3}.M^{2})/n^{2}
≈ - 10^{+47}.10^{-125}
≈ - 10^{-72} Joules
E_{-n} ≈ - 10^{-72}/n^{2} J

so that (m.M) ≈ 10

Thus using expression (48) for this and the heavier examples :

E = -Gm/c

in which the lighter object disappears.

**ACKNOWLEDGEMENTS**

Many thanks are due to many friends and colleagues, particularly Michael Partridge, who examined early drafts of this work and explained to me some of errors therein.

**REFERENCES**

Abramowitz, M., and Stegun, I.A. (eds.), 1972,

"Handbook of Mathematical Functions",
Dover, New York, 9th Printing, p. 504.

Allen, C.W., 1964,

"Astrophysical Quantities", Athlone Press, London, 2nd Edition.

Behr, B.B., 2007.

"Universe", in Volume 19, "Encyclopedia of Science and Technology",
McGraw-Hill, New York, 10th Edition, pp. 80-89.

De Volp, A., 2007.

"Hydrogen Bomb", in Volume 8, "Encyclopedia of Science and
Technology", McGraw-Hill, New York, 10th Edition, pp. 712-713.

Fry, E.S., 2001.

"Neutron", in Volume 14, "World Book", World Book, Chicago, 2001
Edition, p. 155.

Houston, W.V, 1959,

"Principles of Quantum Mechanics", Dover, New York, pp. 58-82.

Schwartz, H.M., 2007,

"Relativistic Mechanics", in Volume 15, "Encyclopedia of Science
and Technology", McGraw-Hill, New York, 10th Edition, pp. 330-332.

**APPENDIX 1**

Equations defining Intermediate Variables

B equation (11)

C equation (12)

F equation (14)

H equation (31)

j equation (29)

k equation (18)

n equation (32)

P equation (26)

R equation (17)

Q equation (19)

s equation (15)

v equation (20)

W equation (17)

y equation (27)

ψ equation (7)