The first major milestone along the path to writing a book with a grandiose title like: A Concise Theory of Truly Everything, is believing in one’s capacity to do so. My societal identifier says I am an American Untouchable; I was born in a Chicago housing project unit, and raised there along with three older half-siblings, by a single mother and her mother; on top of which, I started out believing my “singularly unenlightened state” identified me as one of the least qualified people in the world to write such a book. Accordingly, I had much to overcome to reach this milestone.
My final step toward an awareness that I could make notable contributions to our world’s objective existential knowledge, left the ground on a high school inspiration and landed in college with my first mathematical proof. Be aware that explaining this proof will require me to use basic algebraic and trigonometric operations. If this fact intimidates you out of any curiosity about the technical details of my proof, when you see the first equation, you can just scroll down through the QED and the italicized demonstration of its validity thereafter.
In my junior year at Mendel Catholic Preparatory High School, I was fortunate enough to have one of the top high school science teachers in the state as my physics instructor. Fr. N’s catchphrase was “Physics is Phun”, and he was genially determined to shepherd his charges to that realization.
It was Fr. N, who formally introduced me to Einstein’s theory of special relativity, the counterintuitive principles of which triggered a pleasant lightheadedness within me reminiscent of the feeling I experienced the first time I asked The Question that resurrected my existential curiosity from the tomb of my Catholicism.
For those unfamiliar with the theory, special relativity asserts that the greater the speed at which any physical phenomenon is observed to be displacing through space, the more slowly time will pass for it, as measured by an at rest observer, and the more the observer will experience the phenomenon’s direction of motion length as having shrunk.
According to the theory, these time dilating and length contracting effects are barely measurable in the relatively slow-moving phenomena we typically experience around us. However, they become increasingly apparent as a given phenomenon’s observed rate of displacement through space approaches the speed of light—about 186,282 miles per second. A conclusion of this theory is that time no longer passes for phenomena observed to be displacing through space at the speed to light, and they have no direction of motion length, as measured by their at rest observers.
Another principle of special relativity is that the greater the speed at which a physical phenomenon is observed to be displacing through space, the more energy will be required to further increase its speed. In this context, an infinite amount of energy would be required to accelerate an at rest phenomenon to the speed of light. Accordingly, light embodies the ultimate spatial speed limit.
These intuition-defying assertions made my teenaged head spin. Since I had learned to enjoy this feeling of existential disorientation—thanks to a childhood spent uncovering and repeatedly asking myself questions that triggered it—I devoted significant mental cycles to contemplating special relativity, seeking its underlying logic, and the commensurate expansion of my intuition to encompass it.
Serendipitously, just prior to being introduced to special relativity I had studied trigonometry, and emerged with a solid grasp of the core sinusoidal functions: sine, cosine, and tangent. My subsequent immersion into the depths of special relativity, illuminated by my newly acquired trigonometric knowledge, led to an inspiration.
My inspired idea was that a physical phenomenon’s observed rate of spatial displacement, which can asymptotically approach, but never exceed, the speed of light could be related to how increasing either acute angle of a right triangle will result in its sine asymptotically approaching, but never exceeding, the length of the hypotenuse.
My expanding intuition suggested I was onto something, despite the fact that the standard special relativity equations are not sinusoidal. However, at that time I could not complete my analysis of this insight, because it came to me just as I plunged into the focus-absorbing college application process.
I emerged with a full-ride college scholarship that required me to study engineering; thanks to my good fortune of being in the right place (Mendel Catholic Prep) at the right time (after one of our graduates from each of the two prior years had earned it) to come to the attention of Dr. Nathaniel Thomas. At the time, he was in charge of minority recruitment at the Illinois Institute of Technology, and could recommend promising students for the Bell Laboratories Engineering Scholarship Program. Subsequently, Nate became my first and only role model.
Predictably, while studying at IIT, I ran into my old friend special relativity. Fortunately, during this encounter I was finally free to finish fleshing out my high school inspiration. With my even stronger grasp of the requisite algebraic and trigonometric concepts, I was able to go from inspired idea to mathematical proof.
To understand my proof, first note that the standard (Lorentz) equation for Time Dilation is:
Where:
- Δt°: a given temporal duration measured in an at rest reference frame
- Δt: the same duration measured in a moving reference frame by an at rest observer
- v: the velocity at which the moving reference frame is measured as displacing spatially
- c: the speed of light
Note that where v=0, the rate of passage of time is unchanged; and where v=c, Δt=∞ for any at rest duration, which translates to an at rest observer measuring time as no longer passing for any phenomenon it experiences displacing through space at the speed of light.
While the standard Length Contraction equation is:
Where:
- L°: the direction of motion length of a given physical phenomenon measured while it was at rest
- L: the direction of motion length of that same phenomenon measured while it is displacing spatially, by an at rest observer
- v: the velocity at which the moving phenomenon is measured as displacing spatially
- c: the speed of light
Note that where v=0, the direction of motion length is unchanged; and where v=c, L=0, which represents the direction of motion length measured by an at rest observer of any phenomenon it experiences displacing through space at the speed of light.
In this context, the Lorentz factor gamma, is defined as: γ = 
Accordingly, these special relativity equations can be reformulated as:
Δt = Δt° ÷ 1/γ Gamma Time Dilation
and
L = L° × 1/γ Gamma Length Contraction
The trigonometric formulation of my high school inspiration was:
1/γ = cos(sin-1(v/c)) Inspiration Equation
The expression on the right-hand side of this equation is the key to this interpretation. Understanding it may require this short tutorial on the sine and cosine functions.
Figure 1
For a given right triangle (Figure 1), the cosine of either acute angle (e.g., <BCA) is equal to the length of the adjacent side [BC] divided by the length of the hypotenuse [AC], while the sine of that angle (again focusing on <BCA) is equal to the length of the opposite side [BA] divided by the length of the hypotenuse [AC].
Where θ is either acute angle of a right triangle:
-
-
- cos(θ) = Length of the Adjacent Side ÷ Length of the Hypotenuse
- sin(θ) = Length of the Opposite Side ÷ Length of the Hypotenuse
-
Conversely, the inverse cosine (or cos-1) of the length of the adjacent side [BC] divided by the length of the hypotenuse [AC] is equal to <BCA, as is the inverse sine (or sin-1) of length of the opposite side [BA] divided by the length of the hypotenuse [AC].
Where θ is either acute angle of a right triangle:
-
-
- θ = cos-1(Length of the Adjacent Side ÷ Length of the Hypotenuse)
- θ = sin-1(Length of the Opposite Side ÷ Length of the Hypotenuse)
-
In Figure 1, the length of the adjacent side [BC] of angle <BCA also represents the projection of the hypotenuse [AC] onto the axis aligned with the base of the encompassing right triangle (conventionally the x-axis), while the length of the opposite side [BA] represents the projection of the hypotenuse [AC] onto the axis aligned with its height (conventionally the y-axis). These projections represent the dimensional components of the hypotenuse as a vector. A vector is any quantity defined by both a magnitude and a direction (e.g., the wind coming out of the west at five miles per hours).
Projecting this thinking into space and time—per my high school inspiration—imagine that the x-axis is time, in which the length of [BC] represents the observed rate of temporal displacement of a given physical phenomenon (P). Among the mutually perpendicular height, width, and depth spatial dimensions—all of which are orthogonal to time—the y-axis is the direction in which an observed phenomenon is moving, and so the length of [BA] represents the phenomenon’s observed rate of spatial displacement. In this context, the length of the hypotenuse [AC] represents the phenomenon’s physical displacement vector. Note that the dimensional components of this vector can span both space (as [BA]) and time (as [BC]), using the speed of light (c) as the unit conversion factor.
A key principle of my formulation of special relativity is its assertion that every material phenomenon has an physical displacement vector that, when oriented parallel to its observer’s time axis, identifies the phenomenon as being spatially at rest and temporally displacing at the maximum observable rate of one second per observer second. Where its physical displacement vector is perpendicular to its observer’s time axis, the phenomenon is experienced as moving through space at the speed of light, and no longer displacing temporally. In general, any phenomenon’s observed spatial movement reflects the degree to which its physical displacement vector diverges from a parallel alignment with its observer’s time axis.
Returning to the spacetime interpretation of Figure 1, where P is experienced as spatially at rest, there is no spatial component of its physical displacement vector (the length of [BA]=0), which means it is parallel to the observer’s (O) time axis. Accordingly, each at rest observer is experiencing time passing for P at the maximum observable temporal rate of one second for each second that passes for it (Δt/Δto=1).
Where O experiences P moving through space, O’s measurement of the spatial component of P’s physical displacement vector (the length of [BA]) increases (v→c), as its measurement of the temporal component (the length of [BC]) decreases (Δt/Δto→0). This is the trigonometric essence of time dilation.
This dynamic represents O experiencing P’s physical displacement vector having rotated from a purely temporal orientation (Δt/Δto=1) into an increasingly spatial alignment, whose maximum limit is the speed of light (v→c). Where O measures P as spatially displacing at the speed of light, there is no temporal component of P’s physical displacement vector (the length of [BC]=0), meaning O will measure time as no longer passing for P (Δt/Δto=0)—corresponding to complete time dilation.
Where P’s physical displacement vector is experienced as having rotated out of a parallel alignment with O’s time axis, the direction of motion spatial dimension of P has correspondingly rotated out of O’s three-dimensional material space, and toward a spatially unobservable, increasingly parallel alignment with O’s time axis. The effect of this rotation is that O will experience P’s direction of motion length contracting (L→0), as its velocity increases (v→c), and its time dilates (Δt/Δto→0), hypothetically in agreement with the predictions of special relativity. Now, we can step through my proof of this agreement.
First, let us return to the trigonometric formulation of my original inspiration:
1/γ = cos(sin-1(v/c)) Inspiration Equation
Note that the expression v/c is scientifically designated β, which transforms this equation into:
1/γ = cos(sin-1β) [Step 1a]
Recall that γ = , and therefore 1/γ = 
Applying the β designation to 1/γ in Step 1a, we get:
(1 – β2)½ = cos(sin-1β) [Step 1b]
Squaring both sides:
(1 – β2)½ x 2 → 1 – β2 = cos2(sin-1β) [Step 2]
A fundamental trigonometric identity asserts that:
sin2θ + cos2θ = 1
Subtracting sin2θ from both sides:
cos2θ = 1 – sin2θ
Substituting the equivalent of cos2θ (where θ=sin-1β) in Step 2:
1 – β2 = 1 – sin2(sin-1β) [Step 3]
Subtracting 1 from both sides and then multiplying both sides by -1:
β2 = sin2(sin-1β) [Step 4]
Taking the square root of both sides:
β = sin(sin-1β) [Step 5]
Since sin() and sin-1() are the inverse of each other, they naturally cancel each other out leaving:
β = β Q.E.D.
To demonstrate this proof, consider the example of a physical phenomenon observed as displacing through space at one-half the speed of light. Recall the standard Length Contraction equation is:
Where v = ½c: L = L° × (1 – (¼))½ = L° × (¾)½ = L° × 0.866
The trigonometric Length Contraction equation is:
L = L° × cos(sin-1(v/c)).
Where v = ½c: L = L° × cos(sin-1(½)) = L° × cos(30°) = L° × 0.866
Both formulations assert that for any physical phenomenon observed as displacing through space at one-half the speed of light, its direction of motion length has contracted to 86.6% of its value when at rest.
This proof established the validity of my trigonometric reformulation of special relativity. After deriving it, I spent several months searching for any indication that some academic had already done so, and came up empty.
Additionally, since this trigonometric interpretation would have been conceptually easier for high school students to grasp—and renders Einstein’s subsequent theory of general relativity more intuitively accessible—it seemed odd that they didn’t teach us this version first.
Eventually, I concluded that they probably had not known of this formulation, although even if they had, I still managed to derive it on my own. Consequently, the fact that “an American Untouchable, born in a Chicago housing project and raised by a single mother and her mother” independently uncovered this intuition-enhancing, existential insight, represented my first major milestone toward writing A Concise Theory of Truly Everything.