Based on what has been presented to this point, we may want to examine whether there are any consequences to the way we formulated the equations. In the previous section, we spent some time discussing the equations:

*E = γmc*

^{2}and

*E = hc/λ*

One question that arose is whether or not these two equations can be solved simultaneously. Keep in mind that the first equation is one that we derived for Relativistic Kinetic Energy, whereas the second is the equation that we use in Quantum Mechanics as the definition of the energy of a given photon with a particular wavelength.

Remember that photons are defined as massless particles, hence the reason that the equation

*E = mc*does not apply to the photon. By solving the two equations above simultaneously, it seems that we would be violating a law of physics. However, can we solve these two equations simultaneously without implying that photons have mass? Let us simply mention that both equations are expressions of energy, and that the terms of each (Joules) are equivalent. Also, we can define the energy of any particle that travels with wave properties, regardless of whether the particle has mass or not, as E = hc/λ. It may now be reasonable to give the reader a

^{2}*short*lesson involving a topic from the field of Quantum Theory.

One of the major components of Quantum Theory is the Principle of Uncertainty. This principle states that we can determine either the position of a particle or its momentum, but not both at the same time, with

*any*measure of certainty. If a particle is at rest its position can be determined with ease, since its momentum is zero (0). However, as soon as the particle is given motion, we can no longer say anything about its position or momentum without a certain level of uncertainty.

It is common to use an example of an electron (we will use the symbol

*e*to represent an electron) when describing this principle. An electron is an elementary particle with mass equal to

When an electron is in motion, it may be possible to determine the energy in the given electron with the equation

*E = hc/λ*where λ is the wavelength associated with the electron with uncertain position and momentum. Thus it may still be possible to solve simultaneously the equations E = fmc2(M) and E = hc/ without having to make any unnecessary assumptions.

In the last paragraph of Chapter 4 we solved the two equations simultaneously for the term

*h*, or Planck's Constant. This yielded the equation

h = mc(M)

where

*h*is Planck's Constant and

*c*is the value of the velocity of light in free space. Note also that

*m*would be the mass of the particle in question and

M = |Mr - Mc|

is the physical difference in the length of space for a particle at rest versus the length of space while the particle is in motion (experiencing the wave properties of four dimensional space). Since we know the values for

*h*and for

*c*, let us solve for the unknowns only, and express the above equation as:

h/c = m(M)

2.21 x 10-42 J s2/m = m(M)

2.21 x 10-42 N s2 = m(M)

2.21 x 10-42 kg * m = m(M).

The equations above have had their respective terms substituted to easily visualize the equivalences contained therein. To note,

*h*is in terms of

*J*s*,

*c*is in terms of

*m/s*,

*J*=

*N*m*, and

*N = kg *m/s2*. If we know the mass of the object in question, we can easily solve for the distance that space is 'folded' due to the existing Kinetic Energy. If we use the electron mentioned above and solve for the change in the length of space, we find that

M = 2.43 x 10-12 m

for the mass of an electron,

*e*, mentioned earlier. The less mass that an object has, the greater that space is 'folded.' Since photons contain no mass, space would be as 'folded' as is necessary for the photon to perform its assigned duty. Photons with higher wavelengths would fold space more than photons with shorter wavelengths. This is associated with the red-shift and blue-shift astronomers see when observing distant stellar objects.

We could easily use this relation to determine how much space will 'fold' for a space probe like Voyager I or II traveling to Jupiter or the rim of our solar system. It will be a minimal change, especially since each probe weighs in the hundreds to thousands of kilograms (kg), but it may help explain the mysterious accelerations found in the NASA deep space probes!

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