An Upper Limit to the Electromagnetic Spectrum - Gravitational Collapse of Photons

By Ivars Vilums - August, 1990 (originally published in my "Underware Labs" web site in 1997)

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:

The relationship [Equation 1]

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 [Equation 2] [Equation 3].

Substituting the expressions of  and  for G and c and using kg as the unit of mass we can thus express  as [Equation 4]

which reduces to [Equation 5].

By definition [Equation 6].

Thus, by substituting the expression in  for N,  can be expressed as [Equation 7]

which, through cancellation, reduces to [Equation 8].

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 [Equation 9]

and the frequency of an electromagnetic wave is related to its wavelength by the relationship [Equation 10].

Plank's constant, h, has been measured at [Equation 11].

Substituting the expressions in ,  and  for c, h, v and using units, we can express  above as [Equation 12].

By canceling m and s and simplifying we derive the relationship [Equation 13].

Deriving Mass as a function of wavelength:

Given that energy as a function of mass is expressed by the relationship [Equation 14],

if we substitute the expression in  above for E and the expression in  above for c, we can restate  as [Equation 15]

which can be simplified to [Equation 16].

Since, by definition, [Equation 17]

and [Equation 18],

by substituting the expression in  for N in  above, we can state that [Equation 19]

and substituting this expression for J in  above, we obtain [Equation 20]

which, by canceling m2 and s2 reduces to [Equation 21].

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 [Equation 22]

which can be simplified to [Equation 23].

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 [Equation 24].

Finally, solving for we find that [Equation 25] .

Thus, electromagnetic waves with wavelengths shorter than 2.5619 * 10-34 meters will undergo gravitational collapse.

Symbols used:

c = speed of light
E = Energy (measured in Joules)
h = Planck's constant
G = Gravitational Constant
M = mass (measured in kilograms)
m = meters
N = Newtons
kg = kilograms
s = seconds
v = frequency
l = wavelength (measured in meters)

Note: units of measurement are used explicitly.

Questions and further inquiry:

Does this imply that a collapsed photon can never be observed to reach the speed of light because of the time dilation effect?
Would the hysteresis in interactions appear as inertia?
How is this impacted by the wave shape?
What about the fact that a wave has a positive and negative electrical and magnetic component? Does this imply a "pair production" of black holes? Would they have opposite electric charge? Would they collapse together? Would one event appear to follow the other?
Does this suggest a "Blue Shift" in light as the frequency goes up?
If not, then does this imply a change in the speed of light as wavelength decreases?
How long would this black hole last until it dissipated?
Are they all around and how can we detect them?