_{1}

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Space is a common entity in any static or dynamic system of objects whether the system is a bound system or an open system. Space is described by either measured or abstract dimensions. All motions, momentum and energy transfers take place in space in any given system of objects. By understanding space we understand many physical phenomena happening in space. One method is to observe similarities among phenomena, deduce possible relations and validate the relations through known results. De Broglie wavelength for matter waves is a typical theorized abstract dimension existence of which is established through experiments later. In this paper, the author studies two bound spaces in two bound systems, namely, atomic bound systems and gravitationally bound systems. Both these bound systems have similar characteristics; they have inertial masses in motion with constant kinetic energies for a given orbital distance around the respective central object. In atomic bound space, the central object is the central positive charge which plays the role of creating the bound space around it. In gravitationally bound space, it is the central mass that plays the same role. Thus for these two bound systems a common constitutive relation between the energies of inertial masses in kinetic state, their distance from central object could be present. By noticing the similarities of the two systems, the author proposes such a relation through introduction of an additional space dimension. The existence of the proposed additional dimension is proved in this paper by considering hydrogen atom for atomic bound space and by considering any gravitational system for gravitationally bound space. Though the magnitude of the additional space dimension is different in both the situations, the additional space dimension exists. It is observed that in hydrogen atom the additional space dimension is a constant for the given positive charge of hydrogen atom when electron is in any energy state having principal quantum number of any value from 1 to 5 and with the orbital quantum number zero. For other quantum numbers additional space dimension exists. In the case of gravitational bound space, the additional space dimension is constant for a given mass of the central object for any energy of orbiting inertial mass. The author concludes that total mass energy of an inertial mass having a constant kinetic energy in any bound space is related to an additional space dimension defined by the constitutional property of the central object creating that bound space. As the relation is generic, it throws opportunity to examine other known similar macro, micro or quantum bound spaces created by central objects with different constitutive properties.

Einstein’s mass energy equivalence derived from special relativity [

Let us consider a stable bound system with a central object. Let an inertial mass having rest mass

where

where

For the above bound system, the author proposes

where

We further assume that

First, we consider the quantum bound system i.e. the atomic system, where the central object is electrically charged nucleus. Let us consider an electron with rest mass,

distance

The kinetic energy of electron depends on the quantum number

In (8)

where

where

The values of

We find that

Second, we consider any gravitationally bound space. In gravitational system, as we know the central object is a gravitating mass. We define the central object has a gravitating mass

The kinetic energy of any the planet from Newton’s gravitational force law is given by

where

Substituting for

Thus for a gravitational system the additional space dimension is a constant and is influenced by the mass of central object creating the bound space.

The additional dimension

The author gratefully acknowledges the permission granted by the management of BGR Energy Systems Limited, Chennai, India, to publish this work.

T. V. B. S. Satyanarayana Murthy, (2015) Additional Space Dimension in Bound Spaces Created by a Central Object. Journal of Applied Mathematics and Physics,03,130-133. doi: 10.4236/jamp.2015.32019