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Conservative Flow and the Continuity of World-Lines
John Denker

## *   Contents

1  Conservative Flow in General
2  Warning: Conflicting Terminology
3  Conservative Flow in Spacetime
4  Conservation versus Constancy
5  Quantifying Conservative Flow
5.1  Differential Version
5.2  Integral Version
6  References

## 1  Conservative Flow in General

When energy flows from place to place, it obeys a local conservation law. By that we mean something very specific:

Any decrease in the amount of energy in a given region of space must be exactly balanced by a simultaneous increase in the amount of energy in an adjacent region of space.

We can express the same idea in the form of an equation:

 change in energy = net flow of energy (inside boundary) (inward minus outward across boundary)
(1)

The word “flow” in this expression has a very precise technical meaning, closely corresponding to one of the meanings it has in everyday life. For instance, loosely speaking, water exhibits conservative flow. To a good approximation, water is not created or destroyed; it just flows from one region to another. Any decrease in one region is exactly balanced by a simultaneous increase in an adjacent region.

(By way of contrast, if you want an example of nonconservative flow, consider “the flow of ideas”. You can give me an idea without decreasing your own stock of ideas. But even in this nonconservative and somewhat metaphorical case, flow implies spreading from a neighboring region.)

To repeat: energy cannot change arbitrarily; only certain types of changes are allowed. The relationship between the broad concept of “change” and the more-precise concept of “conservative flow” can be summarized as follows:

 Conservative flow = change + balance + simultaneity + adjacency              (2)

The use of the world “simultaneity" causes some 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. We get into trouble with relativity when we talk about simultaneity at a distance, but since the changes here are adjacent (zero distance) there is no problem with simultaneity.

Our local conservation law is much more useful than the corresponding non-local conservation law would be, for the following reason: suppose I were to observe that some energy has vanished from my laboratory. It would do me no good to have a global law that asserts that a corresponding amount of energy has appeared “somewhere” else in the universe. There is no way of checking that assertion, so I would not know and not care whether energy was being globally conserved. Also there is no way to reconcile a non-local law with the requirements of special relativity.1 See reference 1 for more on this.

To summarize: The laws of physics do not permit energy to disappear now and reappear later. Similarly the laws do not permit energy to disappear from here and reappear at some distant place. Energy is conserved right here, right now.

Indeed, the spacetime view of local conservation is extremely elegant, informative, and easy to visualize ... as we shall see in section 3.

It is important to distinguish the notion of conservation from the notion of constancy, as discussed in section 4.

## 2  Warning: Conflicting Terminology

Beware: The term “conservation” has two wildly different meanings.

 The physics notion of conservation is the topic of this document. It is synonymous with conservative flow. It is defined by expressions of the form equation 1. The vernacular notion of conservation refers to protecting something against destruction, or especially against waste.

 Example: Conservation of energy means energy is always conserved ... automatically, strictly, and locally, in accordance with equation 1. Example: Conservation of endangered wildlife. People get to decide whether they want to conserve this-or-that. Sometimes they decide yes, sometimes no.

While we’re on the subject: Beware: There are two different notions of energy. There is a vernacular notion of energy that is important and widely used, yet is very different from the physics energy.

 The physics energy includes all forms of energy, whether useful or not. The vernacular energy refers to some ill-defined notion of “useful energy” or “available energy”.

 Example: The rest energy is given by E=mc2. Examples: The energy industry. The Department of Energy.

## 3  Conservative Flow in Spacetime

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 two ± T faces in addition to the six 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-line of Stationary Parcel

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 constant, i.e. 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: World-line of Non-Stationary Parcel

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=1
+         +
+      *  +
+     /   +
+    /    +
++++/++++++     t=0
/
/
/    <-- world-line of non-conserved object
```
Figure 3: World-line of Non-Conserved Object

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: Flow with Replacement

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.

Another case that needs to be considered combines the previous idea of non-conservation with the idea of replacement, as shown in figure 5. You can see that the number of objects in the region of interest is constant, because every time one of them is destroyed it is replaced.

```                ++/++++++++     t=2
+/        +
/    *    +
/+    |    +
/ +   /     +
/  ++ /     ++     t=1
/   + /       +
+/        +
/    *    +
/+    |    +
/ ++++/++++++     t=0
/     /
/
/    <-- world-line of non-conserved object
```
Figure 3: Non-Conservative Flow with Replacement

A real-life example of this concerns the number of working light-bulbs in a room. The number of working light bulbs is not a conserved quantity, but the number is approximately constant, because every time a light bulb is destroyed you replace it.

Another example along the same lines is a continuous-flow reactor in a chemical plant, where chemicals of type A flow in and chemicals of type B flow out.

## 4  Conservation versus Constancy

It is important to distinguish the notion of constancy from the notion of conservation:

• When we say some quantity X is constant, or equivalently unchanging, it means simply that the amount of X in the region does not change. There are various ways this could happen, as shown in figure 1 and figure 4.
• When we say some quantity Y is conserved, it means that the amount of X does not change except by flowing across the boundary of the region. This includes the unchanging examples just mentioned, and also includes the changing-but-conserved example shown in figure 2.

To be explicit: Conservation does not imply constancy, nor vice versa. Change does not imply non-conservation, nor vice versa. All four of the possibilities have already been illustrated:

 constant non-constant conserved Figure 1 (static) Figure 2 Figure 4 (flowing) non-conserved Figure 5 (flowing) Figure 3

As touched upon section 1, if you integrate the local conservation law over all space, you can derive a global conservation law as a corollary. It is true that global conservation is the same as global constancy ... but that does not mean that conservation is the came as constancy in general.

## 5  Quantifying Conservative Flow

Now that we have a firm qualitative idea of conservative flow, let us now make the idea completely quantitative.

### 5.1  Differential Version

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

 ∂ ∂t
(energy density) = − div(energy flux)              (3)

or

 ∂ ∂t
(energy density) + ∇ · (energy flux) = 0              (4)

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

Homework: Check equation 4 using dimensional analysis. It shouldn’t take very long!

The structure of equation 4 is very elegant. It has the form

 ∂ ∂t
(...) +
 ∂ ∂x
(...) +
 ∂ ∂y
(...) +
 ∂ ∂z
(...) = ...              (5)

which looks a whole lot like a four-dimensional divergence. If you leave off the ∂/∂t 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. The four-vector consists of the three components of energy flux plus a fourth component, representing the “flux” of energy flowing in the “time direction” – which is just the energy density. This field is divergence-free in D=4.

Equation 4 is frame-independent. Each of the terms on the LHS is frame-dependent, but the LHS as a whole is the same in all frames, the same for all observers. The RHS is zero, which is manifestly invariant.

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 has zero divergence. The energy world-lines are endless.

In the preceding paragraph, all the D=3 statements are frame-dependent, but the D=4 statements are frame-independent.

Of course, energy is by no means the only thing that behaves this way. There are lots of conserved quantities.

• Energy is conserved. Relativistically speaking, mass is a form of energy.
• In the nonrelativistic limit, mass and energy are separately conserved.
• The X, Y, and Z components of momentum are separately conserved. This can be used to derive a useful equation of motion for fluids, namely the Euler equation; see reference 2.
• Charge is conserved.
• Lepton number is conserved.
• Etc. etc.

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

The formulation in this section applies at a single point, or perhaps in the infinitesimal neighborhood of a point. (A neighborhood is necessary to make derivatives meaningful.) It involved density and flux, in particular the four-dimensional divergence of the [energy,momentum] 4-vector. Similar laws apply to the [charge-density,current-density] 4-vector.

### 5.2  Integral Version

Rather than considering a single point, we now consider a region.

We take the differential version (as discussed in the previous section) and integrate it over a region. Using Stokes’s law, we find that the time-rate-of-change of energy inside the boundary is equal to the flow of energy across the boundary.

The integral formulation implies the differential formulation and vice versa, subject to mild restrictions. Obviously things must be sufficiently differentiable, and the region of interest must be simply connected (no wormholes).

## 6  References

1.
John Denker,
“Modern Thermodynamics”
./thermo/

2.
John Denker,
“A Non-Sneaky Derivation of Euler’s Equation”
http://www.av8n.com/physics/euler-flow.htm

1
In some special cases, such as Wheeler/Feynman absorber theory, it is possible to make sense of non-local laws, provided we have a non-local conservation law plus a lot of additional information. Such theories are unconventional and very advanced, far beyond the scope of this document.
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