# Pure Energy?

People often talk about “pure energy” in rather an informal way. In truth there is no proper notion of pure energy. Loosely one often means photons when the term pure energy is used. For example, you may come across statements like: when matter and antimatter collide they annihilate producing pure energy.

Energy is a property of “stuff”; that is a physical system. A configuration of a physical system will have a property that we can indirectly measure, which we call energy. One cannot have energy as some independent “thing”.

As an analogy, you may talk about the colour of a car. Lets say a red car. Being red is a property of the car. One cannot talk about “red” as some notion independent of the objects we see as red in colour. Red does not exist by itself.

So, what is energy?

Informally energy is understood as a property of one physical system that allows it to preform work on another physical system. In essence this means that energy is the property associated with movement or change. It is the “doing” property.

Not that the above tell you what energy is. However, this is not really a problem as physics tends not to deal with the metaphysical notion of existence and what is. Physics deals with mathematically modelling nature. With that in mind, one should keep close the quote (I paraphrase) I believe is due to Feynman:

Energy is a number we can calculate at different points in time and find the same value.

That is energy is something we can calculate, given some configuration of a physical system and (given some technical stuff) we see that the energy does not change in time. That is, it is conserved.

A little more mathematically

We have a very powerful and beautiful theorem that relates symmetry and conservation laws.

Noether’s first theorem: Any differentiable symmetry of the action of a physical system has a corresponding conserved charge.

This theorem is at the heart of modern physics and is based on the calculus of variations.

What does this theorem mean?

Theorems as theorems are by their very nature technical. But we can informally understand some consequences of this statement quite easily.

If the mathematical description of the physical system does not alter upon changes in time then there is a conserved quantity that we call energy.

This is as close to answering the question what is energy? as you can really get. Energy is the quantity that is associated with a physical system not explicitly depending on time.

The caveat here is that the physical system not depend explicitly on time. This is generally reasonable. From a physical perspective this seems natural, any experimental outcome should not depend on when you preform the experiment. Because you get the same result today as you will tomorrow, energy is conserved.

Back to pure energy

I hope I have explained that the notion of “pure energy” is not well founded. Energy is a number that is associated with physics not changing on when you preform your experiments.

Noether’s first theorem makes this association with time and energy explicit. Other common conserved quantities exist:

 Symmetry Conserved Quantity Translations in Time Energy Translations in Space Linear Momentum Rotations in Space Angular Momentum

In the same way nobody talks about “pure angular momentum” as some thing in its own right, no one should use the term “pure energy”.

# The trampoline of gravity

 courtesy of gifmania.co.uk Analogies are an important part of understanding, as well as the popularisation of physics. However, analogies are analogies and at some point always fail to capture the full picture of what is going on. More than that, taking analogies too seriously can lead to misunderstandings.

Einstein’s general relativity basically tells us that massive objects bend the space-time they are sat in and that this is the origin of gravity. To really understand this one has to pull apart the Einstein field equations in all their tensorial beauty. I won’t do that here and now.

 A common analogy here is that of a heavy bowling ball placed on a trampoline. The bowling ball deforms the elastic trampoline surface, it sags, and this is similar to how a massive object, say a star, bends the space-time around it.

One can now “model” photons or test particles by using light balls, say ping-pong balls. The point is that these near weightless balls will not deform the trampoline’s elastic surface. When the bowling ball is not on the trampoline the light balls move in straight lines when given a light initial push. When the bowling ball is on the trampoline the light balls no longer follow straight lines, but curved paths. These light balls are attracted to the bowling ball: thus we have gravity!

This is a great analogy for light rays or photons in general relativity. Light is bent around massive objects like stars. If you have access to a trampoline and some heavy and light balls, play around and experiment for yourself.

However, This analogy seems to be the principle source of misunderstandings and even scepticism of general relativity for the untrained.

Conceptionally the analogy breaks down because the trampoline does not represent the three dimensional space we inhabit, or rather a time slice of our four dimensional world. All we have is an embedding of a two dimensional geometry in our three dimensional flat world.

The trouble is that the space-time of general relativity does not require any such embedding in a higher dimensional flat space. It is of course true that mathematically we can always find (isometric) embeddings in higher dimensional spaces of the geometries found in general relativity, but this does not imply that nature uses such things.

The other issue is that the trampoline analogy really misses the curvature of space-time and only highlights space curvature. The ping-pong balls move about the sheet being “attracted” to the bowling ball because of the depression in the elastic sheet. The trouble is that in general relativity test particles, say photons, move in the “straightest possible path” in four dimensions, including time. This fact is missed by the analogy.

So however useful this analogy is, and I say it is useful, it cannot really describe general relativity. Objections, philosophically or otherwise to general relativity cannot be founded on the trampoline analogy.

 The great man himself. Moreover, there is plenty of direct and indirect experimental verification that general relativity is a good model of gravity. This fact seems rather inescapable: there are no consistent repeatable experiments that, taking into account the domain of applicability and experimental errors, that suggest that general relativity is not a good model. I may say more about this another time.

In short, love analogies, use analogies, tell other people about analogies, however remember they are analogies and no replacement for mathematical models.