Energy definition in special relativity I'm going through the early homework assignments for my special relativity course and I've got myself a little confused about energy. I've got a basic understanding of what the 4-momentum is, having defined it as $m\dfrac{dx^{\mu}}{d\tau}$, and shown that this is equal to $m\gamma (|\vec{v} |)(c,\vec{v})$ where $\vec{v}$ is the classical velocity in the inertial reference frame associated to the above Cartesian coordinates.
Now one of my assignments has began by saying

Denote the components of $p^{\mu}$ as $(E/c, \vec{p})$...

I don't understand why the time component of the 4-vector is being denoted as $E/c$. You can't just go arbitrarily denoting quantities by other quantities! So I'm left wondering whether this is a definition of relativistic energy, or if it follows from some other result in special relativity?
Thanks for any clarification.
 A: Your book may be treating things a little backwards from the way they are usually done. The usual way is to define the momentum four-vector as the combination $(E/c, \vec{p})$, where $E$ is already known to be the total energy (the thing that reduces to $mc^2 + \frac{1}{2}mv^2$ for $v\ll c$) and then go on to show that it satisfies the properties expected of a four-vector. But it sounds like your book defines the four-momentum via $m\frac{\mathrm{d}x^\mu}{\mathrm{d}\tau}$, so you know it satisfies the properties expected of a four-vector from the start, and then they're going to have you prove that the time component of this four-vector has a low-velocity limit of $mc^2 + \frac{1}{2}mv^2$.
They're probably using the notation $E/c$ to be suggestive, but at this stage, it's just an arbitrary notation - that is, they don't intend $E$ to mean energy yet. If you prefer, you could use $p_t$ or something instead, until you actually do show that it's equal to the total energy divided by $c$.
A: 
I don't understand why the time component of the 4-vector [ $m~\gamma(|\vec v|)~(c, \vec v)$ ] is being denoted as $E/c$. 

So the underlying question is two-fold:
Why is "energy" considered the time component of some 4-vector at all?, and
Why this specific time component expression, among time components of all different 4-vectors imaginable?
(Where we're obviously referring to "energy of something which is characterized by $m$", "with respect to the system or reference frame which determined the value $|\vec v|$ of that something".)
A suitably general and readily applicable definition of (how to measure) "energy" seems to be as "time component of the generator of translations"
(or "generator of succession"; alongside the definition of how to measure corresponding space components, namely of momentum as "generator of translations"):
$$\hat E :\simeq \frac{d}{dt}\!\!\big[ ~ \big].$$
Applying this operator to $\tau(\vec v)$ (to what else?) yields (by my naive calculation):
$$\hat E\big[ \tau(\vec v) \big] :\simeq \frac{d}{dt}\!\!\big[ t~\sqrt{ 1 - \left( \frac{|\vec v|}{c} \right)^2 } \big] := \frac{d}{dt}\!\!\big[ t~\sqrt{ 1 - \left( \frac{|\vec x|}{c~t} \right)^2 } \big] = $$ $$ = \sqrt{ 1 - \left( \frac{|\vec x|}{c~t} \right)^2 } - \frac{\frac{t}{(-t^3)}~\left(\frac{|\vec x|}{c^2}\right)^2}{\sqrt{ 1 - \left( \frac{|\vec x|}{c~t} \right)^2 } } = \frac{1}{\sqrt{ 1 - \left( \frac{|\vec x|}{c~t} \right)^2 } } = \frac{1}{\sqrt{ 1 - \left( \frac{|\vec v|}{c} \right)^2 } },$$
where obviously $|\vec v| := |\vec x| / t$.
A similar excercise with one component of the momentum operator $\hat p_x :\simeq \frac{d}{dx}\!\!\big[ ~ \big]$ results in:
$$\hat p_x\!\big[ \tau(\vec v) \big] :\simeq \frac{d}{dt}\!\!\big[ t~\sqrt{ 1 - \left( \frac{|\vec v|}{c} \right)^2 } \big] := \frac{d}{dt}\!\!\big[ t~\sqrt{ 1 - \frac{x^2 + y^2 + z^2}{(c~t)^2} } \big] = $$ $$ = -\frac{t~x}{(c~t)^2} \frac{1}{\sqrt{ 1 - \left( \frac{x^2 + y^2 + z^2}{c~t} \right)^2 } } = \frac{-x}{t~c^2} \frac{1}{\sqrt{ 1 - \left( \frac{x^2 + y^2 + z^2}{c~t} \right)^2 } } = -\frac{v_x}{c^2} \frac{1}{\sqrt{ 1 - \left( \frac{|\vec v|}{c} \right)^2 } )}, $$
with $x$, $y$, $z$ denoting distances in three orthogonal directions, in a flat space, of course.
With suitable proportionality constants
$$\hat E := m~c^2~ \frac{d}{dt}\!\!\big[ ~ \big]$$ and
$$\hat p_x := m~c^2~ \frac{d}{dx}\!\!\big[ ~ \big], \,\,\, \hat p_y := m~c^2~ \frac{d}{dy}\!\!\big[ ~ \big], \,\,\, \hat p_z := m~c^2~ \frac{d}{dz}\!\!\big[ ~ \big]$$
then together 
$$\left( \frac{1}{c^2} (\hat E)^2 - (\hat p_x)^2 - (\hat p_y)^2 - (\hat p_z)^2 \right)\!\big[ \tau(\vec v) \big] = (m~c)^2 $$
which is a result evidently independent of $\vec v$, therefore an invariant characteristic of the "something" whose energy and momentum components were being determined; and $(\frac{E}{c}, \vec p)$ is a corresponding 4-vector expression.
All this applies in the simplest case that the "something" which is characterized by the invariant $m$ is "free". If instead a "potential" enters the consideration then the invariant is rather expressed as
$$\left( \frac{1}{c^2} (\hat E - q~A_t)^2 - (\hat p_x - q~A_x)^2 - (\hat p_y - q~A_y)^2 - (\hat p_z - q~A_z)^2 \right)\!\big[ \tau(\vec v) \big] = (m~c)^2, $$
where $\mathbf A := (\frac{A_t}{c}, \vec A)$ is a suitable 4-vector potential (whose components may in turn be expressed as derivatives of a suitable "phase function" $\alpha( \mathbf x )$), and $q$ represents a "charge".
A: Without calculus, Special Relativity is treated in the fully discrete theory of "causal sets."  Einstein suggested that a discrete theory of space-time could provide an inherent metric, while a continuum requires that a metric be imposed as an accessory to space-time.  In causal set theory, the manifold is formulated in terms of time alone, primitive spatial relations being excluded from the theory.  (If successful, that would constitute the first reduction of parameters in physics since Newton's original reduction.)  I noticed, in a causal set diagram of 3 arrows, that frequency ratios are formed, useful for defining energy ratios in accord with Planck's E=hf.  The "causal link," or discrete temporal transition, is implicated as the quantum of energy ratios, or simply the quantum of energy.  We have here a structural definition of energy and its quantum, in terms of time alone, illustrated in the simplest case by a time diagram of 3 arrows.  Thus we have the prospect of reducing space, time, and energy to causal sets, which are simply formations generated by sheer temporal succession.  See "Causal Set Theory and the Origin of Mass-ratio." http://vixra.org/pdf/1006.0070v1.pdf
