Tuesday, June 18, 2013

Deconstructing Time part 7 - Mach's Principle

Before we introduce Mach or Relativity, let's step back for a moment and recap what we've covered so far. We began with a Use Case - how do we find coordinates in space-time? This is a Use Case that could be applied to Astronomy, to GPS scenarios, to writing Science Fiction stories or movies and potentially many to other situations.  We explained how we would take an outsider's perspective using what is a essentially an IT methodology for problem solving. Then we began introducing key concepts:

  1. Time is inherently connected to motion.
  2. Time seems to be constructed of dynamic events.
  3. Those events exist within frames of reference - those frames define the parameters for simultaneity for the events within. 
  4. Motion occurs (and effects time) not just in our frames of reference but in a cascading chain of ever smaller and ever larger frames - from the subatomic to the universal scale.
  5. The universe seems to be comprised of matter, energy and the space in between these elements in motion. Light is not matter but shares wave behavior with matter. Energy, such as light, can be also broken into individual particles just as matter can. 

All of this is a preface to the more complex theories and concepts we're about to explore.  We still perhaps don't understand what time is other than through the abstract event / frame paradigm that's been presented. This is a good point to ask some questions about time again:

  • Is it a physical construct or a behavioral manifestation resulting from physical structures?
  • Is it merely a human-perceived tool for measuring experience and the physical world around us?
  • Is it an absolute, fixed constant of some sort or is it flexible? (if it is a constant is tied to other constants?)
  • Lastly, if we were to view events as "time particles," can we say that time exhibits wave behavior?

The final question is interesting in that we might then be able to visualize time in a more robust sense - as spreading ripples or waves rather than straight linear progression (the metaphor Stephen Hawking used was 'Time's Arrows'). We'll come back to this later.

Ernst Mach - better know for defining the speed limit for your jet...
What is Mach's Principle? Well, it was just a handful of pages dedicated to explaining more of an observation or question rather than a theory. The principle goes something like this:
In his book The Science of Mechanics (1893), Ernst Mach put forth the idea that it did not make sense to speak of the acceleration of a mass relative to absolute space. Rather, one would do better to speak of acceleration relative to the distant stars. What this implies is that the inertia of a body here is influenced by matter far distant. A very general statement of Mach's principle is "Local physical laws are determined by the large-scale structure of the universe." 
Here's another description:
Mach’s Principle assumes, among other things, that “a particle’s inertia is due to some (unfortunately unspecified) interaction of that particle with all the other masses in the universe; the local standards of nonacceleration are determined by some average of the motions of all the masses in the universe, [and] all that matters in mechanics is the relative motion of all the masses.” Furthermore, “Mach’s Principle actually implies that not only gravity but all physics should be formulated without reference to preferred inertial frames (artificially defined motion contexts).  It advocates nothing less than the total relativity of physics.  As a result, it even implies interactions between inertia and electromagnetism.”
At this point we also need to introduce another new concept - "Action at a Distance" (sometimes referred to as 'spooky action').
In physics, action at a distance is the nonlocal interaction of objects that are separated in space. This means, potentially, that particles here on Earth could interact with other particles across the galaxy or universe without being constrained by the speed of light. This is also what drives Quantum Entanglement.
Confused yet? no worries - so is everyone else...
So, Mach's Principle effectively opened the door both to Relativity and Quantum physics. How does it relate to our discussion though? As we will see shortly, it introduced the notion of relativity to time as well as to motion. More importantly, though it begins to highlight how in a complex systems view of the universe that there are multiple competing variables that must be considered when calculating exact measures of space-time. Let's say we're talking about a Science Fiction story that needs to come up with a reasonable explanation for how teleportation works. Anytime a character is beamed up to a ship or down to planet there would need to vastly superior ability to locate space-time. With today's military grade version of GPS when can get with inches or centimeters of locating a coordinate. However that's not good enough, two inches too low might mean our character would have his feet "materialize" within solid rock. More exact measurements, down to the micron (one millionth of a meter) level would be needed to make the technology safe (and it would have to be able to compensate for scenarios where the target ground is uneven etc.).

Now it's time to define Relativity, we'll start with Special Relativity:
Einstein's theory of special relativity is fundamentally a theory of measurement. He qualified the theory as "special" because it refers only to uniform velocities (meaning to objects either at rest or moving at a constant speed). In formulating his theory, Einstein dismissed the concept of the "ether," and with it the "idea of absolute rest." Prior to the generation of Einstein's theory of special relativity, physicists had understood motion to occur against a backdrop of absolute rest (the "ether"), with this backdrop acting as a reference point for all motion. In dismissing the concept of this backdrop, Einstein called for a reconsideration of all motion. According to his theory, all motion is relative and every concept that incorporates space and time must be considered in relative terms. This means that there is no constant point of reference against which to measure motion. Measurement of motion is never absolute, but relative to a given position in space and time. 
Special Relativity is based on two key principles:

  • The principle of relativity: The laws of physics don’t change, even for objects moving in inertial (constant speed) frames of reference.
  • The principle of the speed of light: The speed of light is the same for all observers, regardless of their motion relative to the light source. (Physicists write this speed using the symbol C.)

Contrary to popular belief, the formula E=MC squared was not published as part of Einstein's Theory of Special Relativity, but was instead published the same year (1905) in a separate paper (Einstein published 4 papers that year).  Of particular interest to Einstein in his development of Special Relativity was the ability to apply a coordinate system to space-time (this will sound familiar as we've already introduced these concepts prior but outside the discussion of Relativity):
An event is a given place at a given time. Einstein, and others, suggested that we should think of space and time as a single entity called space-time. An event is a point p in space-time. To keep track of events we label each by four numbers: p = (t,x,y,z), where t represents the time coordinate and x, y and z represent the space coordinates (assuming a Cartesian coordinate system).
Even physicists have a sense of humor 
Now let's define General Relativity...
There was a serious problem with Special Relativity however, it was artificially constrained on several levels; the most important of which is the fact that it doesn't address acceleration (it handles reference frames at rest or at constant speeds only). The other constraint was the speed of light - C. Einstein has initially sought to help unify his theory with Maxwell's theory of electromagnetism (wave theory) - Maxwell had set C as a constant (that couldn't be surpassed) and Einstein carried that through. However, there was already evidence that this wasn't entirely accurate. The force responsible for the most obvious violations of C and the force responsible in many cases for acceleration was gravity... 
It (General Relativity) states that all physical laws can be formulated so as to be valid for any observer, regardless of the observer's motion. Consequently, due to the equivalence of acceleration and gravitation, in an accelerated reference frame, observations are equivalent to those in a uniform gravitational field.
This led Einstein to redefine the concept of space itself. In contrast to the Euclidean space in which Newton’s laws apply, he proposed that space itself might be curved. The curvature of space, or better space-time, is due to massive objects in it, such as the sun, which warp space around their gravitational centre. In such a space, the motion of objects can be described in terms of geometry rather than in terms of external forces. For example, a planet orbiting the Sun can be thought of as moving along a "straight" trajectory in a curved space that is bent around the Sun.
The most important concept from our perspective in both of these two theories is time dilation (which is referred to as Gravitational time dilation in General Relativity:
Gravitational time dilation is an actual difference of elapsed time between two events as measured by observers differently situated from gravitational masses, in regions of different gravitational potential. The lower the gravitational potential (the closer the clock is to the source of gravitation), the more slowly time passes. Albert Einstein originally predicted this effect in his theory of Special Relativity and it has since been confirmed by tests of general relativity. 
Another way to think of time dilation is this, the faster you travel (now were dealing acceleration), the slower your time progresses as opposed to those you left behind at home.  For science fiction fans, one of the few movies that remain true to this otherwise plot-busting feature of modern physics is the Planet of the Apes. In that plot, astronauts leaving Earth around 1973 travel at near the speed of light for 18 months and end up back on Earth 2,000 years later. Of course, it is unlikely that we'd experience the 'damn dirty ape paradox' if we attempted such a trip.

Apes don't kill apes but they will take over if we leave for a few thousand years...
So, let's recap again what this all means:

  1. Several incredible and somewhat intuitive insights, transformed modern physics starting just over a century ago with Mach's Principle. (this isn't to say that other folks like Maxwell didn't provide brilliant insights, but for our investigation Mach's insight was particularly important)
  2. This led to the determination that the universe behaves certain common laws yet those laws support relative perspectives of outcomes.
  3. The notion of space-time as both a coordinate system and as a curved (multi-dimensional) geometry emerged. 
  4. Because of all of this (and more) it was determined that our perception of time is different based upon what reference frame we inhabit. The differences in relative temporal perception are mostly due to the nature of the motion involved (for one or many participants involved).   
  5. Gravity, which was a mysterious force before now becomes part of space-time itself under General Relativity and interestingly also seems to exhibit wave behavior. 

It is often said that General Relativity is the last classical theory of Physics as it is the last major one that doesn't involve Quantum mechanics. Relativity takes us much further down the path towards explaining time - but it stops short of doing so adequately.

In our next post, we'll point out why Relativity falls short and introduce some of the foundational concepts of Quantum Mechanics.

copyright 2013, Stephen  Lahanas


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