Quote:
Originally Posted by modest  You’re right and I’m an idiot for that strawman.
I clearly forgot there aren’t any textbooks that tell me why or how mass curves spacetime. Why stuff makes the universe present itself as non-Euclidian. Why or how the postulates of GR are correct or much in the way of conceptualism behind them.
I suppose it’s an often-asked question that is just as often attacked. People would say it’s a philosophical question or that it isn’t part of science. Maybe they’d say our human minds are incapable of understanding or conceptualizing what the math tells us. Or, they may just as well say someone is an ass for asking. Whatever the response, it never seems to be a straight-forward answer to the question. For my part in that, I apologize and offer the best answer I have:
I don’t know or understand the physical mechanism behind gravity. I don’t know if the answer can be found in cosmology so I obviously don’t know if you’re going about it the right way or not.
The next best thing I can offer is something I think you may have already read: The Ontology and Cosmology of Non-Euclidean Geometry
I will quote the conclusion for anyone who hasn’t:
-modest |
Very good posts
modest. Once again, thanks to your intervention, this thread (as others before) is becoming interesting.
I was originally going to post more examples of gravitating systems that exhibit a geometrical strucutre consistent with Lagrangian mechanics (and there are many): to show that there is a definite relation between all bounded systems (regardless of scale or complexity). And, that this relationship leads to a deaper understanding of the concept of spacetime curvature - whereby there is an intrinsic competition between objects (the maxima of potential) and 'empty' space (the minima of potential), via the gravitational field.
I will continue to post these, as the discussion necessitates (particularly those related to barred galaxy structure). For now though, let me get right to the heart of the debate: the elucidation of the physical mechanism behind the gravitational interaction.
To understand what is happening in the field (something which we cannot see) we have to look at (not only) how objects (or a test particle move) within the combined fields of massive bodies, but too, where they eventually find their equilibrium position (where they end up 'at rest'). That was the purpose of presenting examples of systems observed in nature.
If we accept that many gravitating systems display a propensity to collect, group, coalesce, arrange themselves, in a pattern consistent with that which is observed locally (in the solar system), i.e., that both general relativity and Lagrangian dynamics are operational ubiquitously, then a striking conclusion can be drawn: There exists in the nature of gravitating systems (and thus there exists in Nature) a (I'm going to get a cup of coffee...) fundamental property inherent in the fabric of space responsible for the competition between massive bodies and the underlying space within which they are immersed, resulting in the curvature of spacetime: a tension or distortion (in, or of) the manifold. This tension (or curvature), juxtaposed between objects (the maxima) and points of equilibrium, certain Lagrange points in the potential (the minima), is/are distributed in such a way that allows systems to remain stable (for whatever timescale), or at least allows objects to remain in stable orbits (e.g., at L4 or L5 zones), thus forming patterns observed at all scales. This propensity, possibly coupled with a 'natural selection' process during the initial formation of such systems, and the resulting velocities requirments attained for objects to remain in orbit, are the reason why celestial bodies do not end up in one great massive fireball: i.e., tend to “fall down” as Newton wrote, “into the middle of the whole space and there compose one great spherical mass.”
Let's examine how this interplay might work.
Spacetime curvature (gravity) must be treated as a deviation or departure from linearity, i.e., the deviation occurs away from linearity, from a flat, Euclidean, Minkowski spacetime and in accord with GR. All gravity (curvature), whether hyperbolic or spherical in geometric form (whether a compression or a stretching of spacetime), is considered a departure from the standard zero condition. For convenience, let us consider gravity a positive departure from linearity.
What we have is an absolute scale for gravity (spacetime curvature) that begins at absolute zero, flat, Euclidean spacetime, and becomes increasingly curved as the gravitational potential increases. Euclidean spacetime, however ideal that state described as gravity-free, where a test particle introduced into the field will experience no net acceleration, i.e., it will feel no force.
We set thus a lower limit on the gravitational field curvature for the spacetime manifold, and in doing so a reveal a basic property (or two) of spacetime: There exists a fundamental limit inherent in nature that manifests itself as field-free space. Beyond that limit, even in principle, spacetime cannot be curved, i.e., there is no ‘beyond’ that limit (just as there are no negative temperatures on the Kelvin scale). Mathematicians invented the concept of ‘field’ to articulate how a specific quantity might vary from point to point in space. What we are saying here is that there exists a fundamental boundary at the state where the field is reduced to zero, at critical points, when interacting fields cancel out. This value of zero for curvature, though not the absolute zero value of gravitational potential energy at infinity, is a relative potential (where fields cancel to zero, i.e., at the inner Lagrange saddle point), meaning that a zero curvature can exist in a wide variety of potentials and in all systems where two or more interacting fields cancel (which by definition is in all N-body system), resulting in a net force of zero.
Terminolgy: A global minimum value for gravitational potential energy (PE) is also a local minimum value (or point), i.e., there is no value less than absolute zero (the value at infinity). On the other hand, a local minimum value (consistent with the value of PE at certain Lagrange points) is not necessarily equal to the global minimum value. (Respectively, local and global are synonymous with relative and absolute). For the purpose of this discussion, the fact remains, the local value of gravitational potential at L1 is zero, the gravitational gradient vector vanishes.
Here is an example of a saddle point (See
Critical Points of Functions of two variables):
The result of this interaction and reduction of potential in the combined fields is that there are smooth peaks (minima potentials) and troughs (or wells) the depth of which depends on the relationship between the mass-density of the object(s) inside the well, and the minima reference frame. So far, so good. The metric properties of spacetime adapt to accommodate mass, creating or inducing a dynamic stress or surface tension on the manifold: always in relation to the local minima (which is always zero).
So gravity is not caused by mass-energy. Spacetime is distorted (compressed, not stretched, as we will see later) in the presence of mass-energy as a result of tension against flat (otherwise 'empty') space. Spacetime curvature (gravity) is a property of the four-dimensional vacuum surface, just as surface tension is a property of the surface of a liquid. In the case of surface tension, the liquid is not being stretched downward (say, when an insect walks or glides on the surface of water), since water tends to ...
The physical behavior of the gravitational field (the deviation from linearity) in the presence of massive objects cannot be understood without taking into consideration the tension or stress created on the original surface (the vacuum manifold itself - like the surface of water before being disturbed). This 4-D surface, along with its associated tension in the presence of mass-energy, governs not only the shape objects can assume when immersed in the vacuum, or the degree of contact a massive body can make with another body, but too (and as a result of both the former and latter), the geometric structure and dynamics of the entire system (the placement or location of objects in relation to others, the ability and propensity to attain orbital velocity required for maintaining equilibrium, and possibly too the threshold or maximum density allowable for a given area of point: more on this later).
Applying general relativity and Lagrangian mechanics to the forces and interactions that arise due to 'manifold tension' in the presence of mass-energy accurately predicts the behavior of the system.
The nature of the vacuum ground state is that there is a tendency to minimize potential energy, to minimize its energy state, to minimize its surface area, to remain as flat and as empty as possible.
As a result of potential energy minimization, the vacuum substrate (the 4-dimensional spacetime manifold, or surface) will assume (or tend to assume) the smoothest and flattest shape allowable. When massive objects populate the manifold the tendency toward flatness, linearity, is still present, thus creating stress or tension in the field. Mathematical proof that smooth shapes or flatness minimize surface area can be found with the use of Euler-Lagrange Equations.
Here is another example of saddle points (See
Critical Points of Functions of two variables):
By symmetry, it seems safe to conclude that the local minima (Lagrange points, the local minima of curvature) are at the origin of the gravitational phenomenon, i.e., the physical mechanism of gravity must be associated with this minima, or it must at least be taken as a starting point for the elucidation of the physical mechanism.
It can be shown that gravitating systems display mechanical equilibrium and that this equilibrium is attributable to the interplay between the local maxima and the local minima of potentials: both, in effect canceling each other out. So, going out on a limb, I hypothesize that gravitating systems are in mechanical equilibrium; the sum of the forces, and torque, on each massive body in the system is zero. In another way, GR explains the observed equilibrium without fine-tuning orbital velocities. (Recall, Newtonian mechanics was unable to do so, since gravity was considered an attractive force. The only way to get around this problem would be to consider Lagrangian points repulsive). This means, too, that the effects of GR do not only over-ride Newtonian gravity when velocities are close to c, or when mass-energy density tends to the global maxima. Indeed GR is the theory of choice, as well, when mass and velocities are small.
Here is an example of visible Lagrange points within the context of a galaxy; the infamous ESO 566-24 four-armed barred spiral galaxy, optical image, slightly enhanced by CC (See
here the original photo and simulation): Note the similarity with the above contour plots of maxima and minima potentials.
I will continue to pursue the geometric argument in the next few posts, since it is through this pathway, only, that the physical mechanism of gravity can be understood.
CC
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