Introduction and An A Priori Derivation of the Minkowski Space Time Interval

Yes, I am GREAT SCIENTIST!

Ok, not really.  I am actually not a professional scientist at all (though I hope to be some day).  However, I thought I would start a blog to share my exploits in amateur research.  Indeed, I confess as I currently have no access to actual experimental equipment, my ideas are all of the gandenken experiment kind.  Thus, you will find no sigmas here, unless or until I actually enter a research program.  Sorry to disappoint you, but in actuality, the purpose of this blog is mostly to share interesting mathematical derivations and wild speculations, mostly related to whatever it is that I have decided to focus my fickle mind on at the moment.  Often, that's physics, but other times, you will encounter other topics ranging from number theory and abstract algebra, to proof theory.

My goal in writing this blog is to create a delicate balance between overly simplified arguments you will find in most blogs geared toward laymen and overly technical, jargon, laden arguments that you will find in a scientific paper.  In my experience, though it is often true that the devil is in the details, it is also frequently as true that much of the structure of the devil can be found in a straightforward manner by a competent non-expert once some fundamental details have been explained.  In short, most profound truths spring from a handful of key insights.  (I suspect, however, that there are some profound results that don't adhere to this principle.  The Four Color Theorem comes to mind). With this in mind, the purpose of this blog is not to introduce you to fundamental concepts, but, in an entirely self-indulgent way, to provide you, the reader, with navigational charts of territory I have covered, am currently covering, and will cover, though not all in order.  Now, on to the derivation...

The Minkowski Metric

Today, I will start with physics, specifically, Special Relativity, one of my favorite topics.  It is a beautiful theory, which serves as a rather startling example of how profound, foundation shaking, predictions can be derived from simple, though obscure, postulates.  It is also a theory that requires few to no measuring apparatuses to actually derive well-tested experimental predictions--that is to say, it is not a theory of the phenomenological kind.  This makes it ideal for an aspiring physicist, such as me, to use it as a springboard off of which to learn physics while in the transitional, awkward, period of trying to make enough money to go back to school and become a full-time crackpot--err researcher...yeah...that's what you heard me say...researcher...

This derivation developed out of something I consider a rather guilty pleasure: arguing with people online.  I won't get into details, but there is a well known commentator in the Physorg community that has a rather--shall we say--loose idea of physics.  Specifically, I would describe his brain as an empirical demonstration of Murphy's law, where if there is a well established principle that every competent person understands, his brain will find a way to misinterpret it.  But, I digress.  Over the course of arguing with this person, it became quite clear that he was unable to resolve the notion that two observers could measure a physical result differently without crashing all of physics like it was pretend 23:59:59, 12/31/1999 in his mind--even though we do it all the time in regular old Newtonian mechanics when we change coordinate systems.  Understandably, the addition of conflicting temporal measurements does introduce some conceptual difficulties, but the wonderful thing about SR is that we've had 100+ years to develop all kinds of pedagogical strategies for carefully guiding the physics neophyte through the difficult waters of simultaneity.  Anyway, it occurred to me that it might help to develop less ambiguous description of space-time points, which served as triggers for his mind to jump to completely incorrect conclusions.  This lead me to asking if there was a way to uniquely identify a space-time point independent of, say velocity.

In other words, I asked, is there a way to assign a unique coordinate to a spatial point that didn't move to anyone?  Obviously, the inverse Lorentz Transform would not satisfy this, for everyone, except the person in the rest frame.

Mathematically, we would want to find a function, F, such that

\begin{equation}
F(r,t) = F(L_r(r,t),L_t(r,t))
\end{equation}

where r is the spatial position, t is the temporal coordinate, L_r is the Lorentz transform of the spatial position, and L_t is the Lorentz transform of the temporal coordinate.  In English, F of  (r, t) is the same applied to (r', t'), where (r', t') is the transformed coordinate.

This kind of equation, where two sides of the equation involve F applied to different arguments, is known as a functional equation.  Functional equations hold a special place in my heart for a number of reasons I will not get into, at least not now.  They are also notoriously difficult to solve.

Now, one general strategy (if not the general strategy short of guessing) for solving awful equations is to try to convert them to more familiar equations that have already been solved.  In this case, I wondered what the differential properties of this equation would be.  It turns out that this equation is ideal for asking that kind of question.  To see this, it suffices to solve only the 1+1 dimensional version of this equation.

In this case, we have
\begin{equation}
F(x,t) =F((x-vt)\gamma,(t-\frac{xv}{c^2})\gamma)
\end{equation}

Now, it's a well known fact that as v approaches 0, the Lorentz factor approaches 1.  Hence, we can choose v to set the Lorentz transformed point arbitrarily close to the original.  This seems to be what we are looking for.
In particular, let's replace v with a differential.  Let's call it dv.

Now, we have
\begin{equation}
F(x,t) =F((x-dvt)\gamma,(t-\frac{xdv}{c^2})\gamma)
\end{equation}

Using the chain rule, and straightforward, but tedious algebra, we can now turn this into a partial differential equation:
\begin{equation}
F(x,t) =F(x,t) - \frac{\partial F}{\partial x}(x,t)t\mathrm{d}v - \frac{\partial F}{\partial t}(x,t)\frac{x}{c^2}\mathrm{d}v
\end{equation}

or

\begin{equation}
\frac{\partial F}{\partial x}(x,t)t =  -\frac{\partial F}{\partial t}(x,t)x/c^2
\end{equation}

One thing that is immediately clear, if you know a little vector calculus, is that this is saying that

\begin{equation}
\nabla F \cdot \begin{pmatrix} t \\ \frac{x}{c^2} \end{pmatrix} = 0
\end{equation}

However, I won't use this.  Instead, I will find curvilinear coordinates that can be used to separate the variables.  In particular, I will find a parametrized family of curves such that F is constant along each curve.  If I can find this family, I will have solved the equation.  Specifically, we start with F(x,y) and completely differentiate it to get

\begin{equation}
dF(x,t) = \frac{\partial F}{\partial x}dx + \frac{\partial F}{\partial t}dt
\end{equation}

and set dF(x,t) to 0 so that
\begin{equation}
\frac{\partial F}{\partial x}dx + \frac{\partial F}{\partial t}dt = 0
\end{equation}

Plugging (5) into (8), we see that
\begin{equation}
-\frac{\partial F}{\partial t}\frac{x}{tc^2}dx + \frac{\partial F}{\partial t}dt = 0
\end{equation}

So,
\begin{equation}
xdx = c^2tdt
\end{equation}

Integrating both sides,
\begin{equation}
x^2 = c^2t^2
\end{equation}

Look familiar?  We're not quite done yet, though.  What we have found is that F must be constant along the curve defined by x² - (ct)² = 0 .

In fact, I lied.  That's not the only solution from integrating both sides.  The general solution is
\begin{equation}
x^2 = c^2t^2 + k^2
\end{equation}

where k is an arbitrary constant.  Thus, we have found our family of parametrized curves, k being the parameter.  F would be constant along each of these.

Indeed,
\begin{equation}
F(x,t) = F(\sqrt{x^2 - c^2t^2},0 \ \mathrm{s})
\end{equation}

It follows that if we define an arbitrary function, H, such that
Indeed,
\begin{equation}
F(\sqrt{x^2 - c^2t^2},0 \ \mathrm{s}) = H(\sqrt{x^2 - c^2t^2})
\end{equation}

Then, the general solution of (1) is
\begin{equation}
F(x,t) = H(\sqrt{x^2 - c^2t^2})
\end{equation}

--That is assuming that F is differentiable.  Obviously, one function that satisfies this equation is then the Minkowski metric.  Thus, not only have we inadvertently derived the Minkowski metric in 1+1 dimensions, but we have found that every single possible differential function must also essentially be the Minkowski metric, too.

Conclusions

There are some interesting things to note here.  First, I seem to have shown that the answer to my initial question is negative.  We cannot uniquely identify a spatial point without, essentially, fixing our origin on something physical that moves.  We must either make ourselves the origin of the universe, or concede that origin to someone else who is not at rest with us.  You'll notice, also, that I did not do a higher dimensional derivation, for the simple fact that I actually haven't yet.  Based on the method outlined above, it seems like we might be able to get away with deriving results far more interesting than that which I obtained for 1+1 dimensions.  In particular, at least naively, it doesn't seem like there isn't any good reason why we couldn't construct a function that is vector valued, and not just complex valued.  Finally, I wonder if we can extend this method to solving a larger class of functional equations.  Anyway, that's all for now.  Signing off.

[Update:
Some things have been brought to my attention.  First, this probably should read Minkowski Space-Time Interval, not metric.  Second, I didn't actually derive this interval for every single pair of points, but only for pairs of points where one is the origin.  However, I am almost certain a similar argument can be made for those space-time intervals too.  If and when I get around to it, I will post the full argument.  Mea culpa.  I am sure the frequency of me saying that will increase as the number of people who know better than I do who read this blog increases.  Much appreciation to the guy who pointed these things out.  I am not sure if he wishes to be named.]