(formerly "Inertial Theory of Gravity ", modified)
This is an attempt to explain fundamental reasons for gravity, going beyond its relativistic definition as spacetime curvature. The “Big Spin” model suggests that gravity is a result of rotation of the Universe’s hypersphere (not a Gödel's rotation). It assumes that space possesses elastic properties and Newton’s law of inertia holds true for higher dimension(s). This model also suggests that space curvature, along with the “hyper-” rotation of the Universe, replaces relativistic concept of spacetime curvature, providing similar quantitative results.
Although General Relativity is a widely accepted theory of gravity with some aspects of it proved experimentally, it serves as a mathematical model, rather than physical explanation of a phenomena that can be matched with an intuitive analogy from everyday life. There are also views that consider treatment of time in Relativity philosophically questionable (admittedly, it is inevitable for such a general concept). The ultimate challenge the author sees is to present a rather simple explanation of gravity, which would still agree with the experiments. Here is an attempt to provide such a model.
It is accepted to think of our Universe as a multi-dimensional object. For purposes of this paper, we will narrow our choices to 4D sphere (or “hypersphere”), without considering other possible shapes. I suggest to treat the Universe as an elastic shell (perhaps I should say “hypershell”). We can think of objects of the Universe as “confined” within it, that is being unable to leave the hypershell, but still free to move along it. Although there are ways to speculate about the fine structure of the objects, we will not discuss it in this paper.
I believe it will be a valid assumption that, besides the “Big Bang” expansion, the Universe can also have some other kinds of movement. All of them (including expansion) should be considered with respect to higher dimensions. As it will be shown later, we are particularly interested in the possible rotation of the Universe. I should stress that rotation of a hypersphere is quite different from the rotations we see in everyday life.
First, we are going to consider a couple of scenarios of a moving universe. Let's assume that if a universe moves in n > 3 dimensions along the straight line and with constant speed, it will not affect objects in the universe (generalizing Newton's law of inertia). I would also suggest that if a universe moves with some acceleration, but accelerates in a direction orthogonal to itself, objects in the universe will not “perceive” this acceleration.
Let's look at the lower-dimension analogy.
Suppose that we have a flat two-dimensional universe (Fig.1), which moves with acceleration in a direction perpendicular to it (projection of the vector on the universe's plane is zero). In this case flat “apple” behaves as there is no acceleration at all (some extra details on that in the next chapter).
In the next example (Fig.2) the same universe moves with acceleration oriented in some other (non-perpendicular) direction. Now projection of the acceleration on the universe's plane is non-zero (vector ). This results in apparent movement of the “apple” with acceleration in respect to the universe.
Let us forget for a moment about spacetime curvature of General Relativity. If, as suggested earlier, Universe's “hypershell” possesses elastic properties, than in case of accelerated motion we could expect Universe's objects to bend space (as much as word “bend” applies to four dimensions). It would be somewhat similar to what would happen to a sheet of fabric, if it had a flat massive object on it and moved with acceleration oriented perpendicularly to the surface of the fabric.
Again, let us look at our two-dimensional analogy of the Universe.
A flat universe moves along the perpendicular to it with acceleration . It causes massive objects to “bend” the universe in the direction opposite to the vector of acceleration.
We will discuss possible reasons for the Universe's acceleration a bit later, now we will take a closer look at what happens in vicinity of the massive bodies, where space is curved.
We're taking step forward from our simplified view on Fig.1. Here (Fig.4), we take space bending into account. In vicinity of a massive body (big “apple”) projection of the acceleration vector on the surfaceis non-zero, which causes small objects “slide” towards the massive ones (here we do not consider space bending caused by the small objects themselves and effects caused by that).
Now, what could cause our Universe to constantly accelerate? The easiest way to get a constant acceleration is to use rotational motion. In our case, we should think of rotation that provides same acceleration for all points of the hypersphere. Essentially, we want for every point of the hypersphere the following condition to be satisfied:
Where r is a vector from the center of the hyperspehere to some point of the hypersphere and k is a constant. As well, perhaps, we want it to be a rigid rotation. In 4D space with coordinates (x,y,z,w) it seems like simultaneous rotation in two planes, for instance (x, y) and (z, w), will provide the desired result. Possibly, we could as well think of other kinds of rotation, perhaps less intuitive ones (so far we deal with more than three dimensions).
Let us formulate the above considerations in a single statement (the “Big Spin” model of gravity):
Our Universe is a four-dimensional rotating hypersphere. The rotation creates a centripetal acceleration which is generally orthogonal to the Universe's space in every point and not perceived by the objects of the Universe. However, the centripetal acceleration, along with the elastic reaction of the space, causes curving of the space in vicinity of massive bodies. As a result, in the curved areas the acceleration is not orthogonal to the space, which appears to the Universe's objects as gravity.
Now, you ask, what about time? According to General Relativity, time is the part of the “spacetime curvature” that is responsible for everyday gravity the most. So far time curvature in GR attributes to acceleration, I'm suggesting that time curvature is just a reinterpretation of the centripetal acceleration, varying in vicinity of massive bodies. Note that we still have “space curvature” part of GR untouched, which will continue to explain light bending, etc.
Fig. 5 Cross section of a two-dimensional universe.
Take a look at Fig. 5. It deals with lower dimensional analogy and shows a cross section of a 2D universe, picturing our “sliding” model of gravity (it uses centrifugal force instead of centripetal accelleration, for convenience). We will use this picture as a helper reference for considering a case of a 3D spherical symmetric mass (Massive body). In a polar coordinate system, external to the Universe, we have
Parameter A defined by mass of the body, elastic properties of the Universe and parameters of the Universe's rotation.
Acceleration caused by the rotation:
and can be thought of (if really necessary) as a kind of a “curved time”
It is very attractive to use rotation for explaining gravity, because rotation is a very natural form of motion in the Universe. It is in a way “absolute” and self sufficient - no need for an outside observer or inertial coordinate system. This model also keeps gravitational and inertial masses equal. It also shows why gravity is so weak comparing to the other forces.
Another speculation that the “Big Spin” model of gravity can lead us to is a possible explanation of accelerating expansion of the Universe without a need for “dark energy” (or one can say Universe's rotation is the “dark energy”).
Admittedly, getting rid of time component of spacetime, we need to suggest an alternative explanation for the effects described by Special Relativity. Although such a discussion goes beyond this paper, perhaps revision of the Michelson-Morley experiment can be instrumental in such effort.
See also: considerations on Space Curvature