Spacetime is like a movie reel, or better a stack of pictures on top of one another. Time is just a way of labelling the individual frames or pictures, and the word “move” just means to have different coordinates in space at different times ie to be at different places in different pictures. It doesn’t make sense to ask for the image of something in one picture to be in a different picture.

# Month: July 2024

## Why KE is Quadratic

For any motion with constant acceleration over a time interval, the product of acceleration times distance travelled over that interval is equal is equal to half the change in the square of the speed in the direction of acceleration. (This is just the junior high school version of the calculus identity \int{x’’ dx}=\int{x’’ x’ dt}=\int{x’ x’’ dt}=\int{x’ dx’}=\Delta(x’^2/2).)

So the quantity that is increased by applying force through a distance (to do “work” on a particle) is quadratic in its speed. But why is this important enough to give it a special name? That is because it allows us to define a quantity that is conserved throughout the evolution of any physical system.

As a result of Newton’s law of action and reaction (which is basically just a way of expressing conservation of momentum), in the motion of any system of particles the sum of mv^2/2 for all the particles plus the net work done against outside forces is a constant. We call this the “Energy” of the system and identify the part involving the speeds (that does not include work against the outside world) as the “kinetic” part of that energy – and the outside work (which includes an arbitrary constant depending on what we take as the starting point) is a called “potential” energy since it could in principle be returned to the system in future interactions.

## Our Motion Through Time

Spacetime with matter and gravity is not symmetrical. There is a particular singularity relative to which what we identify as our experience includes information (which we call memories) about events “closer” (not in spatial distance but in the spacetime metric) to the singularity (which we call the “past”), but not about those further away (which we call the “future”). Part of what is in our memory is experiences in which the scope of our memory was smaller. This gives us the feeling of becoming progressively further away from the singularity – ie of moving “forward” in time.

## Statistical vs Mathematical Physics

Mathematical Physics is the study of what can be really proved about our theoretical models of physical systems. This differs from other kinds of theoretical physics because physicists often take the lack of experimental refutation of a mathematically invalid calculation as “proof” that the result is correct.

Statistical physics (aka Statistical Mechanics) is the study of physical systems having so many degrees of freedom that it is not feasible to measure a complete set of the individual observables (such as the positions and momenta of all the molecules in a volume of gas or the angular momenta of all the electrons in a crystal), but for which some observables (such as temperatures and pressures), defined as averages of those most naturally considered as forming a complete set, are expected to evolve in a way that does not depend on the specific values of all the variables needed for a complete description.

Many of the expected behaviours of these averages are assumed by physicists without any complete proof; and one important area of mathematical physics is the filling in of these missing proofs. A classic text of this sort is the book ‘Statistical Mechanics: Rigorous Results’ by David Ruelle.

## What is Horizontal?

*John Platts’s answer to Why does a mirror reverse things horizontally but not vertically?* includes some nice illustrations and discussion but declares that the front-to-back reversal is not horizontal.

Interesting point. But I think we use the word “horizontal” to refer to any part of a line or plane that is perpendicular the line joining it to the Earth’s centre, regardless of whether or not our view of it actually appears parallel to the horizon. We often test for this property by looking at it from a position where it lines up (left-to-right) with the horizon but two people building a house will generally agree that a beam is horizontal even when they are not looking at it from such a position. (If that were not the case the word would be a lot less useful because the property of a beam being horizontal would depend on the observer and would be lost whenever she changes her orientation.)

So, in my understanding, horizontal lines aren’t necessarily *parallel to* the horizon in a perspective view, but (if the world was flat) their infinite extensions (would) all *terminate on* it (though due to the Earth’s curvature that actually happens a tiny bit above what we see as the visible horizon).