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An Upper Limit to the Electromagnetic Spectrum - Gravitational Collapse of Photons
By Ivars Vilums - August, 1990
This paper attempts to show that a sufficiently short electromagnetic wave will undergo gravitational collapse. It will derive a value for the size of this wavelength, and speculate on some of the ramifications of this phenomenon.
It is well known that, for an electromagnetic wave, as the wavelength decreases the energy of the wave increases. It is also well known that energy and mass are equivalent. It has been demonstrated many times that electromagnetic waves are affected by gravitational fields and that electromagnetic waves exhibit the mass equivalent of the energy of their wavelength. It is also accepted that when a mass is concentrated in a sufficiently small enough space, it will undergo gravitational collapse. The radius of this space for any mass is called the Schwarzschild radius.
Starting with an expression for determining the Schwarzschild radius as a function of mass value and an expression of energy as a function of wavelength, we will derive an expression to determine the mass equivalent of any wavelength of electromagnetic energy. Knowing this, we can then derive a relationship that expresses the Schwarzschild radius of any wavelength. Since energy increases as wavelength decreases, we can then solve for the wavelength size that will have a sufficient energy density in a space whose size is equal to the Schwarzschild radius for its equivalent mass.
Expressing the Schwarzschild Radius as a function of Mass:
shows how the Schwarzschild radius for any mass M is related to the gravitational constant G and the speed of light c.
G and c have been measured as
Substituting the expressions of  and  for G and c and using kg as the unit of mass we can thus express  as
which reduces to
Thus, by substituting the expression in  for N,  can be expressed as
which, through cancellation, reduces to
Deriving Energy as a function of wavelength:
It has been shown that the energy of an electromagnetic wave is related to its frequency by the relationship
and the frequency of an electromagnetic wave is related to its wavelength by the relationship
Plank's constant, h, has been measured at
Substituting the expressions in ,  and  for c, h, v and using units, we can express  above as
By canceling m and s and simplifying we derive the relationship
Deriving Mass as a function of wavelength:
Given that energy as a function of mass is expressed by the relationship
if we substitute the expression in  above for E and the expression in  above for c, we can restate  as
which can be simplified to
Since, by definition,
by substituting the expression in  for N in  above, we can state that
and substituting this expression for J in  above, we obtain
which, by canceling m2 and s2 reduces to
Deriving the Schwarzschild Radius as a function of electromagnetic wavelength:
Substituting the expression of  above for M in  above, while taking into account the cancellation of the kg factor of  between steps  and  above, yields the relationship
which can be simplified to
Deriving the Schwarzschild wavelength:
We can now calculate using  the Schwarzschild radius for any wavelength of electromagnetic radiation. Since, as the wavelength gets shorter, the energy in the wave (and thereby also its equivalent mass) increases, it is apparent, then, that there is a wavelength sufficiently short enough that it is shorter than the Schwarzschild radius of the energy in the wave.
The wavelength equal to this size can be computed by modifying  above to take into account the two energy peaks in a wave. The relationship is
Finally, solving for we find that
[Equation 25] .
Thus, electromagnetic waves with wavelengths shorter than 2.5619 * 10-34 meters will undergo gravitational collapse.
Note: units of measurement are used explicitly.
Questions and further inquiry: