*" in indefinitely small spacetime regions in which it is possible such coordinate system acceleration that is not induced any gravitational field, is still valid SR that is the invariance of the term : ds ^ 2 = - (dx1) ^ 2 - (dx2) ^ 2 - (dx3) ^ 2 + (cdt) ^ 2.*

As above all the effects of spacetime curvature between two distinct and not-inertial reference systems, in relative motion one with respect to the other, occur and are observable only from one into the other and viceversa.

In RG gravity force depends on the acceleration induced in a mass by sourronding spacetime curvature generated by another mass.Therefore the free-falling mass weight would an apparent strength of electromagnetic origin (repulsive between electronic clouds) due to the opposition of a surface to such motion. So the weight of a man on Earth would be due to this potential acceleration induced by S - T curvature by Earth center mass. But it was said that curvature occurs only between not-inertial systems moving relative each to the other so in reality, for the same GR, we should not feel any weight. In a stationary reference frame, with respect to the center of mass of the Earth, such as the one fixed to a man with his feet firmly on the ground, there is no kinematics trajectory and, therefore, no first and second derivatives. Acceleration is added, at a later date, on the basis of an alleged (potential) free fall motion. but it is a mere mathematical artifice, and perhaps, applying spherical shelltheorem, law of Newton better explains how gravity develops under Earth's surface and why it is worth zero in the center.

As just said General Relativity describes the gravitational interaction not as a distance action between massive bodies, as it was in the Newtonian theory, but as a result of a physical law that binds and flow distribution in spacetime of mass, energy and momentum with the geometry ( more specifically, with the curvature) of the spacetime itself.

The geometry of spacetime, in particular, determines which are inertial reference systems: are those ones associated with observers in free fall, which move along geodesic trajectories of spacetime. The weight force is, in this way, an apparent force observed in not-inertial references systems.

Equivalence Principle states that

*inertial mass and gravitational mass of a body have the same value*. This equality is an experimental fact that it does not follow from any principle of classical physics ;in fact the roles of these two quantities are very different: the inertial mass measures how the body prevents the application of a force, as stated by the second law of motion, instead gravitational mass measures the ability of a body to attract another mass, according to the law of universal gravitation.

The fact that these two quantities experimentally found to coincide implies the fact, already observed by Galileo around 1590, that the trajectory of a body in free fall is not dependent on the properties of the body.

Einstein issued the following thought experiment. Consider an observer located inside a closed room. If the room is resting on the earth's surface, the observer perceives a downward force due to gravity: throwing a ball into the ground you can measure the size. If the room is instead in space, away from gravitational fields, contained in a rocket accelerating upwards, the observer perceives, in this case also, a downward force: this force, due to the inertia of his body, is the same force that normally is perceived in the departure and arrival in an elevator. The equality results in the following fact: the observer can not in any way to find out whether he feels is due to a gravitational field or an acceleration.

Similarly, if the room is in a free fall to (for example) the Earth, the viewer inside receives no gravity: if he drops a coin notes that this does not fall to the ground but remains suspended in mid-air . The observer has no means to understand if it is in a universe without gravitational fields, or if it is falling towards a planet.

Let’s see the question from a more closely mathematical point of view.

Einstein field equation describes the spacetime curvature depending on the density of matter, energy and pressure, represented by the stress-energy tensor T.

In the left of the equation appears the Ricci curvature tensor that measures
the curvature of a Riemannian manifold that is a differentiable manifold on which are defined the notions of distance,
length, geodesics, area (or volume).

The notion of a differentiable manifold is a generalization of the concept of curve and surface differentiable in arbitrary n dimensions. As well as a differentiable curve is an object which locally resembles a straight line, or a surface that locally resembles a plane, locally an n-dimensional manifold looks to an n-dimensional Euclidean space. The adjective "differentiable" indicates the fact that this local "similarity" guarantees the ability to uniquely associate each point in a "tangent space" of the same size of the manifold (such as a tangent line to a curve, or a tangent plane to a surface).

Minkowski spacetime is the R^4 space with the g yensor summarized in the form :

The notion of a differentiable manifold is a generalization of the concept of curve and surface differentiable in arbitrary n dimensions. As well as a differentiable curve is an object which locally resembles a straight line, or a surface that locally resembles a plane, locally an n-dimensional manifold looks to an n-dimensional Euclidean space. The adjective "differentiable" indicates the fact that this local "similarity" guarantees the ability to uniquely associate each point in a "tangent space" of the same size of the manifold (such as a tangent line to a curve, or a tangent plane to a surface).

Minkowski spacetime is the R^4 space with the g yensor summarized in the form :

ds^2 = -(cdt)^2+dx^2+dy^2+dz^2

The Energy-Momentum tensor, also called Stress Energy Tensor, describes the energy and momentum flow associated with the field.

In General Relativity motion quantity is the four-momentum. In Special Relativity the four- vector is the quadrivectorial generalization of classical mechanics momentum, which is a four-dimensional spacetime vector always tangent to a particle world line, that is tangent to its trajectory.

In lagrangian mechanics, every mechanical system is characterized by a Lagrangian which allows to describe motion system by means of Euler-Lagrange equations, or the Hamilton Principle.

Its importance is fundamental in classical mechanics and in many areas of physics, as it allows to derive the motion equations of the system (excluding the influence of non-conservative forces).

For the variational principle of Hamilton, the temporal evolution of a mechanical system minimizes the action. Its Lagrangian proves to be the difference between the kinetic energy and total potential energy. For this difference the Euler-Lagrange equations is equivalent to the second law of dynamics.

On the right of field equation appears the Sress – Energy Tensor that express the conservation of the four-vector, given by the continuity equation.

Therefore free fall postulate is an artifice designed to enable the prior space - time curvature and apply the mathematical formalism that holds all GR.

To be even more clear: since the derivative of a function (and partial derivatives appear in the mathematical operators of Einstein field equation) is the limit of the incremental quotient tending increment to 0, the postulate, in fact, allows existence of increments on spacetime trajectories, increments that, surely, at least in the case of the potential trajectory of a mass placed on the surface of the earth do not exist with respect to the mass center.

In fact, without gravitational fields space time isn’t curved (it is the Minkowsky’s plane space – time) ; in it can be chosen infinite inertial reference systems and between them are valid Special Relativity and Lorentz transformations. A general curved space – time has a much important property that connects it, as we can say, to the most familiar euclideus plane space. As far as it could be curved, it’s always possible to consider a little portion of it where it's practically plane. We can better understand this concept considering earthly surface. It's a two dimensional space (variety) curved where we can define curvilinear coordinates as latitude and longitude. In great scale we can’t take away the curvature of earthly surface and the effects are well visible to all. Instead for a mason building an house earthly surface is plane and he doesn’t mind of the problem. In every curved space – time it's always possible to choose a reference system respect on which space – time is locally plane and inertial (Minkowski’s space – time).

To do it is sufficient to imagine a body that falls down free in a gravitational field. Respect to this body, for a limited time, the other free bodies falling down with it look to satisfy the inertia law. The ones standing keep standing, the ones in uniform motion get on in this way. Respect to this reference system, falling down for a short time, space – time is the plane Minkowski’s one of SR. The astronauts have experience of this when they are parked on earthly stationary orbits (In effect it is the same thing as they really would fall down freely). In their space craft they test zero gravity. Here on the Earth is possible to verify this for a short time when, for example, an airplane gets an air vacuum or in some games at lunapark.

The fact that space – time is curved by masses that create gravitational field is a concept outside commune experience. In a curved space – time are invalid the rules and the properties of euclideus geometry, that is the geometry of our daily life.

To clarify better this concept let's consider an inertial reference system K and a not inertial reference system K’ in uniform rotation respect to K. Let's consider also a circumference tied with K.

Respect to K the fraction between the circumference and its diameter is π. Respect to K’, that rotates in a clockwise direction, the circumference is seen rotating in the opposite way. Every little segment of the circumference is seen from K’ moving on with v speed. For some instant every little segment by which is made up the circumference is seen contracting respect to K’ in accord with the Lorentz contraction law for which the fraction between circumference and its diameter is, respect to K’, unequal of π (diameter doesn’t be subjected to the Lorentz contraction for the reason that it doesn’t move respect to K’ in the sense of its length).

This simple example shows that space, respect to an accelerated reference system, is not plane but is curved, for the reason that are not valid the rules of euclideus geometry and so, remembering the precedent statement :

*“in physics is real what is measurable",*we could say that time dilatation and lenght contraction are real (or, that is the same thing, absolute) and that since a gravitational field is equivalent to an accelerated reference system, space – time is curved by a gravitational field.

This is the fundamental principle of General Relativity : weight force is due to the curvature induced by masses in the space – time.

In reality space - time is seen curved by k but it does not bend .... About the rigid disk paradox.

In the Hafele - Keating experiment some atomic clocks placed in aircraft flighting around the Earth remained desynchronized than identical clocks on the ground. The reason was that they accumulated delay during linear motion phases, in which worked RS, than those fixed to the ground, and mainteined it in landing when RS was no longer applicable. In MT this way is interpreted with the existence of an absolute reference system for the time. The same reasoning does not apply to the lengths because there is no accumulation phase, but only contraction measured by the system ground reference.

The possible existence of a absolute reference system for the time could be experimentally demonstrated by the presence of not relative effects but intrinsic to a noT-inertial reference system in prolonged free fall motion towards a center mass attractor capable of generating a strong S – T curvature gradient (ex. a neutron star), with a cesium clock which measures time decay of a radioactive atom and would compare it to the theoretical one that would be measured if the reference system standing. Obiouvsly this experiment is impossible.

Ultimately in MT it is conclude that since 'the weight force (more' generically gravity), is not an attractive force and not depends on S - T curvature, it can not 'be than an active pushing force (with application point, modulus and direction),generated by a dynamic and not static gravitational field.

Stefano Gusman

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