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flow, conservation, and continuity of world-lines



When explaining local conservation of energy, I wrote:
Flow = change + balance + simultaneity + adjacency (eq. 1)

The use of the world "simultaneity" caused lots of people to wonder whether
such a statement could be relativistically correct. Well, it turns out
that the local conservation law is 100% kosher, relativistically and
otherwise. Indeed, the spacetime view of local conservation is extremely
elegant and informative and easy to visualize. Let's take a look.

An ordinary box in D=3 space has six faces. We can call them the +-X
faces, the +-Y faces, and the +-Z faces.

In D=4 spacetime, the corresponding object has eight faces. It has +-T
faces in addition to the aforementioned spatial faces.

Now let's draw some spacetime diagrams. For simplicity we will conceal the
Y and Z dimensions, and portray only the +-T and +-X faces of the box.

Here is the world-line of a !stationary! parcel of energy sitting at the
location of our box. The world-line enters on the -T face and exits on the
+T face:

|
|
|
++++|++++++ T=1
+ | +
+ | +
+ | +
+ | +
++++|++++++ T=0
|
|
| <-- world-line

Figure 1

World-lines crossing the +-T faces contribute to the time derivative of the
amount of energy in the box. In the foregoing case, there are two equal
and opposite contributions (one flowing in at T=0, and one flowing out at
T=1). To say the same thing in other words, the amount of energy inside
the +-X walls of the box is the same at T=0 and T=1, so the amount of
energy in that spatial region is not changing with time.


In contrast, here is the world-line of a non-stationary parcel of
energy. The world-line enters on the -T face and exits on the +X face:

/
/
/
+++++++++++ / T=1
+ +/
+ /
+ /+
+ / +
+++++++/+++ T=0
/
/
/ <-- world-line

Figure 2

In this case, the amount of energy in the spatial box at T=1 is less than
the amount of energy at T=0. The energy is not locally constant, but it is
locally conserved. The energy wasn't destroyed, it just flowed across the
+X boundary into the neighboring box.

We will make this more quantitative in a moment.

You can see where this line of reasoning is leading. The following notions
are formally equivalent:
-- Local conservation of energy.
-- Continuity of energy world-lines.

Contrast this with some non-conserved object, such as a photon. Here is
the world-line of a photon that enters the box and gets absorbed; the
world line enters the box but doesn't go out again:

+++++++++++ T=0
+ +
+ X +
+ / +
+ / +
++++/++++++ T=1
/
/
/ <-- world-line of non-conserved photon

Figure 3


To quantify the net flow across the +-X boundaries, we need to know
something about velocity and energy density. Imagine a fluid that has the
same velocity and the same energy-density everywhere. That will give rise
to a situation like the following:


/ /
/ /
/ /
+++/+++++++ / T=1
+ / +/
+/ /
/ /+
/+ / +
/ +++++++/+++ T=0
/ /
/ /
/ / <-- world-line

Figure 4

In this case, the amount of energy in the box is constant, because whenever
an energy parcel leaves via the +X face, some other energy parcel is
entering via the -X face. Contrast this with figure 2, where the energy
flowed out and was not replaced. This non-replacement could occur because
the energy density was lower off to the left, and/or the velocity was lower
off to the left. So we see we need to be concerned with
-- the energy density
-- the velocity
-- the left-to-right variation thereof.

The product of energy density times velocity is called energy flux. What
really matters is the X-derivative of the flux. If there is more than one
spatial direction, the flux is a vector (just like the velocity) with X, Y,
and Z components. What matters is the X-derivative of the X-component of
flux, the Y-derivative of the Y-component, and the Z-derivative of the
Z-component --- in other words, the divergence.

So we can write the local conservation law as
d/dt(energy density) = - div(energy flux) (eq. 2)
or
d/dt(energy density) + del dot (energy flux) = 0 (eq. 3)

This is a 100% formal accurate statement of the local conservation law. It
expresses the continuity of energy-parcel world-lines.

Homework: Check equation (3) using dimensional analysis. Shouldn't take
very long!

The structure of equation (3) is very elegant. It has the form
d/dt(...) + d/dx(...) + d/dy(...) + d/dz(...) = ... (eq. 4)
which looks a whole lot like a four-dimensional divergence. If you leave
off the d/dt term you've got the plain old three-dimensional
divergence. The deal is that (unlike, say, magnetic flux lines) energy
world-lines are not divergence-free in D=3. They are only divergence-free
(i.e. endless) in D=4. The energy world-lines are represented by a
four-vector field consisting of the energy density plus the three
components of energy flux; this field is divergence-free in D=4.

To repeat, in D=3 you can (temporarily) have a whole bunch of energy flux
diverging from some region of space. [You cannot have magnetic flux lines
diverging like this, not even temporarily.] The divergent energy flux will
deplete the energy density in that region. Meanwhile, if you take a D=4
view of the same situation, you will find that the energy density/flux
four-vector is divergence free. The energy world-lines are endless.

Of course, energy is by no means the only thing that behaves this
way. There are lots of conserved quantities.
-- Energy is conserved.
-- The X, Y, and Z components of momentum are conserved.
-- Charge is conserved.
-- Lepton number is conserved.
-- Etc. etc.

The picture of world-line continuity applies to them all.