Chapter 31: The Three Accelerations and Einstein’s Elevator

The fundamental assumption for Dr. Einstein’s  triumphal paper on General Relativity regards the equivalence of gravitational mass and inertial mass. It is so basic, so fundamental that it has come to be regarded as one of the basic building blocks of all Science and reason, and has become one of the most stalwart principles in all physics. It has even been given its own name, unlike the axiom regarding the speed of light. We call it the Equivalence Principle, and it roughly goes like this:

In the far reaches of space the effects of gravity and inertial acceleration are indistinguishable.

Dr. Einstein was never actually able to prove this however, in fact, no proof of this assumption has been yet demonstrated to this day, and there is still some debate among physicists as to whether or not the Equivalence Principle is actually true.  However, it has been quantifiably tested to a high degree of accuracy by folks that had a lot more time on their hands than I do, and if gravitational mass and inertial mass are not the same, the difference is on the order of the 1 X 10-12 parts to one.

In order to ‘prove’  this, the good doctor came up with a novel thought experiment that has since been dubbed ‘Einstein’s Elevator’ and it goes something like this:

Suppose that you are in a sealed elevator, unable to view anything outside of your little box. (We’ll have to assume that it is airtight, too.)  Based on the experiences that you could have, or any experiments that you could perform inside of your elevator, you could not determine whether you were resting on the surface of the Earth, or accelerating smoothly through free space at a rate of 9.81 m/sec2. Notice that this says nothing about velocity, but it basically implies that we’re talking about subluminal velocities here, and as long as we are this statement is basically true, at least as far as we can currently divine. This is roughly illustrated below:

Screen Shot 2014-02-23 at 9.03.34 PM

As you can see, Dr. Einstein is pretty busy reading, and probably wouldn’t be looking out the window even if he had one, to see where he was, and the point is that even if he could, he couldn’t, by any means known to man, tell whether it was gravity or acceleration that was causing him to stick to the floor.

But we’ve just determined that there are actually three distinct velocities, dimensions if you will. And Dr. Einstein’s thought experiment only includes the effects of one type of motion, that is; linear velocity. If these velocities in this new interpretation are separate and unique, they should each have a unique acceleration , with unique effects in order to be distinguished from each other; and in fact, they do. Let’s start with the easiest example, the acceleration associated with rotation.

As it turns out, with a rotational velocity, one does not have to be in an accelerating state to create a gravity-like acceleration since the ‘centrifugal force’ does that quite nicely. One utilizes the centrifugal force every time one uses a salad spinner to dry off salad greens, or when one swings a bucket full of water around, keeping the water still inside. But the most common use for the centrifugal force in in a centrifuge. A centrifuge spins the subject in a highly controlled manner and through the force created by the spinning (aka gravity) can separate heavier things or parts of a mixture from the lighter things in it. This is how medical professionals separate blood, for example. The higher gravity created by the spinning of the tube of a sample of blood in the centrifuge causes the heavier, more dense blood cells to settle to the bottom of the test tube, allowing the lighter lymph fluid to rise to the top. This is not unlike how gravity affects air BTW, and will be discussed at greater length when we talk about hurricanes.

But be that as it might; if we take Einstein’s elevator, attach it to a relatively short rigid arm and rotate it at a constant velocity, we can also create a gravitational field whose maximum intensity, or strongest point, is on the bottom of the elevator, as illustrated below:

Screen Shot 2014-02-23 at 9.04.42 PM

So if we rotate Einstein’s Elevator at a constant rotational velocity around any fixed point, be it the end of the arm that we’ve attached to it, at the top of the elevator, or even about its central axis, we create a gravitational field on the floor. However, this field will not be as uniform, or as ‘flat’ as those created by linear acceleration or a large mass’s gravity. It can therefore be differentiated from them by the observer inside utilizing the object shown in the illustration above; a marble, because the marble will roll to the point that has the strongest gravity; which is what it will do in any given situation.

Screen Shot 2014-02-23 at 9.05.43 PM

From the illustration above, we can see why. The force crated by the motion is proportional to the length of the arm (or length of the radius of rotation). In this case, if the length from the rotation point to the middle of the floor of the elevator is some distance, ‘r’, then the length of the distance to either corner must be longer, in this case, ‘r’ plus an additional length delta, due to the geometry of the model. This means that, while both points are moving with the same rotational velocity, the absolute speed of the speed of the corners is higher than the speed of the center of the floor due to the increased length of the rotational arm, r + delta, and the centrifugal force there must be stronger. Therefore, if one were in an elevator rotating as shown and one placed a marble at the center of the floor, it would roll to either edge, due to the increased acceleration (gravity) at that point.

So in this case, one can tell the difference between the gravity created by a large mass (planet) or linear acceleration and that created by rotation by employing a marble to help to visualize the difference in the strength of the field.

For the case of spiral motion, the situation is a little more complex, but the insight provided by the marble is even more conclusive. Since a spiral velocity can vary in both rotation rate and radius length, simultaneously, there are many possible choices for the motion that we choose to analyze. For this example, we’ll pick one, the logarithmic spiral, and assume that we’re moving in along the curve at a constant velocity. This particular spiral motion pattern will be the best for this thought experiment since the rotation speed (which is increasing) and the radius length (which will be decreasing) will be inversely proportional with regard to their change, and it will be easy to maintain a constant forward speed. This case will allow our observer to experience ‘gravity’ in a direction close to his accustomed orientation, and more importantly, maintains a constant line of motion and velocity, as shown below.

Screen Shot 2014-02-23 at 9.06.55 PM

Since we are changing both the length of the arm and the rate of the rotation as we travel inward, there will be two simultaneous accelerations at work. The shortening of the radius over time will cause an acceleration in the opposite direction of that portion of the velocity, that is, away from the center of the rotation (much like any strictly linear velocity would cause); one that is toward the floor. The angular velocity will also create a force toward the floor, but in this case, since the rotational speed is changing, too, and unlike the case of the pure rotation, this force will not be evenly distributed.

Screen Shot 2014-02-23 at 9.08.08 PM

This complex motion will cause there to be a gravity gradient across the floor that intensifies as it tends toward the trailing edge of the elevator (because the rate of rotation is accelerating), and no matter where our observer sets the marble on the floor in this situation, it will always move to the trailing corner, as shown above. So, the observer in his little elevator, will always know whether he is on a spiral path and can always determine the difference between the acceleration that it creates and gravity, linear motion or pure rotation.

So, two of our velocities, rotation and spiral motion, will both create accelerational (gravitational) fields that can, even in a small, closed environment, be distinguished from each other and from a field created by gravity or linear motion, which is not exactly contrary to what Dr. Einstein claimed, since he was only talking about linear motion. He just didn’t consider those motions in his grand analysis.

And the reasons that these motions each create their own unique gravitational fields has to do with the basic geometries of their respective ‘dimensions’; as we shall soon see.

But wait! There’s more!

Since there are also basic geometrical differences between linear motion and gravity, if one examines a very extreme case, it turns out that the elevator bound observer can tell the difference between them.

In the linear dimension, the gravity created in a closed system undergoing a unidirectional acceleration exhibits itself in the opposite direction of the direction of the acceleration and is pretty much uniformly created on any plane within the closed system that is perpendicular to the direction of the motion. That is not to say that it is only felt on planes that have that orientation, but it is most effectively experienced there due to the direction of the force created. That situation is graphically illustrated below:

Screen Shot 2014-02-23 at 9.09.16 PM

One can see that a marble placed on the floor will have no preferred location since all points on the floor have the same gravitational  potential.

Gravity itself however, is a force created by the presence of a material object, a mass, with some density, and is uniformly exhibited in all directions at once. And since most of the objects that are under consideration are basically round, then, naturally, so are the gravitational fields that they create.

So let us consider a case involving extreme gravity. In this example, as illustrated below, the gravitational field is so small and intense that there are variances in the strength of that field across the floor of the elevator, due to the round geometry of the mass providing the gravity and the flatness of the floor of the elevator, and the intersection of that round field with the flat floor. This geometry means that there is a different distance from each location on the floor to the center of the mass (well, actually the gravitationally equivalent spots appear in pairs in our two dimensional model due to symmetry, but whatever).

Much like the example of the rotational field, the floor of the elevator experiences the gradient of the field. Because the center of the floor of the elevator is closer to the center of the source of the gravity, it will be strongest there and the marble will roll to the very center of the floor and stay there.

Screen Shot 2014-02-23 at 9.11.42 PM

While it is true that this example is stretching things a bit, since gravity that is strong enough to do this would probably be crushing the observer into a puddle on the floor, the point is this: gravity has a positive curvature and the acceleration created by linear movement does not, so an observer, if he could withstand this environment, he would actually be easily able to tell if he were in the presence of gravity or not, because of the way the geometry of the field as it intersects his vehicle.

Does this in any way violate the Equivalence Principle or in any way imply that there is a difference between gravitational mass and inertial mass?

No. In the opinion of the author, inertial mass and gravitational mass are the same because gravitation is created by inertial mass, and therefore, they are just different aspects of the properties exhibited by the phenomenon that we call mass.

But the most important conclusion to draw from of all of this is that each velocity has its own unique acceleration, which, although being similar to gravity in their effects on mass, can be distinguished by the geometry of the field that they create, and therefore, while being basically equivalent, are not truly equal. It is exactly these sorts of details that an examination of this new geometry can coax out from everyday, ordinary experience, and can allow the observer to have deeper insight into the actual interactions and forms that the matter and energy in our universe creates.

In the next chapters, we’ll take a more detailed look at each of these new dimensions and the geometries that are inherent to them in an effort to provide better definition for each and to provide us with the tools to utilize them for some better descriptions of our physical world. This will lead to some surprising conclusions, and change the interpretations of many things that we thought that we knew. Off we go!

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