THE EXPOSÉ OF GRAVITY PAPER I

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6 Oct 2022
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ABSTRACT

Gravity has for long been perceived as a constant accelerating downward force, that at its very core holds and forms the general colloidal mass of objects as particles condense into the organic and inorganic matter they later form. Gravitation is a force manifested by acceleration towards each other of 2 free material particles or bodies or of radiant-energy quanta. Likewise, weight is a consequent result of gravity. In relation to what one of the founding fathers of modern physics, the great Einstein said, I hitherto say thus;
“Everything is not constant, but relative to the momenterality of both space and time both independently and interwovenly as space-time.”
In a broad sense, the fabric of space-time as perceived by Einstein is seen to be the four-dimensional framework of differential manifold on which a metric tensor is imposed that solves most of the mundane field equations such as the Einstein-Field equations, and that metric tensor gives rise to geodesics and objects/bodies that are not experiencing any other force move along the geodesics described by that metric. Gravity as one of the pillars of science amongst four has a peculiar bedeviling conceptuality and I am in a bid to uncover it to its uttermost. My very understanding of gravity is that gravity exists at all points in space and is universally interwoven with time as well. It doesn’t fluctuate at the perceiving range of 9.6 - 10ms-2 but since its universal, the point in the geometry of space-time in which you measure it is crucial because in light of the relativity-idea, the magnitude of gravity due to the unequal distribution of particles thereby matter and so measurement of gravity is not constant but relative. The conceptuality that gravity is universal, I’m obliged to accept but the ideology that the magnitude of gravity is a universal constant, I cannot. In the next few pages, I do not intend to implode the debunkment of the foundation of classical physics. My idealistic view of gravity is;

“Gravity is not a force but as a consequence of masses moving along geodesic lines in a curved space-time caused by the uneven distribution of masses across the universe.”

 

IF GRAVITY ISN’T A FORCE, HOW DOES IT ACCELERATE OBJECTS? (ADVANCED)


Einstein said that there is no such thing as a gravitational force. Mass is not attracting mass over a distance. Instead, it’s curving spacetime. If there’s no force, then how do you explain acceleration due to gravity? Objects should accelerate only when acted upon by a force; otherwise they should maintain a constant velocity. But this doesn’t seem to be the case when observing bodies experiencing free fall. A few of the explanations I’ve found online refer to equivalence and the thought experiment of a man standing on Earth experiencing the same g-force as a man in a rocket being accelerated into space. I understand why those conditions are the same, but I fail to see how that explains a brick falling from a building accelerating at 9.8ms-2. Also, in that thought experiment a force is being exerted i.e. the thrust of the rocket. This is perhaps the most common question about general relativity. If gravity isn't a force, how does it accelerate objects?
 
General relativity says that energy (in the form of mass, light, and whatever other forms it comes in) tells spacetime how to bend, and the bending of spacetime tells that energy how to move. The concept of "gravity" is then that objects are falling along the bending of spacetime. The path that objects follow is called a "geodesic". Let's begin by looking at the bending side of things, and then we'll come back to look at geodesics.
 
The amount of bending that is induced by an object is directly related to that object's energy (typically, the most important part of its energy is its mass energy, but there can be exceptions). The Sun's mass is the biggest contribution to bending in our solar system. So much so, that it dwarfs the bending of spacetime by the Earth to the extent that to a very good approximation, we can just consider the Earth to be mass-less as it travels around the Sun (we call this the test particle limit). Similarly, when you're standing on the Earth, the Earth's mass dominates the bending of spacetime over your own, and so you can treat yourself as a mass-less test particle for all intents and purposes. However, truth be told, you warp the spacetime around you just a teeny tiny bit, and that does have an impact upon the earth in response.
 
Now, let's get back to those geodesics. A body undergoing geodesic motion feels no forces acting upon itself. It is just following what it feels to be a "downward slope through spacetime" (this is how the bending affects the motion of an object). The particular geodesic an object wants to follow is dependent upon its velocity, but perhaps surprisingly, not its mass (unless it is mass-less, in which case its velocity is exactly the speed of light). There are no forces acting upon that body; we say this body is in free-fall. Gravity is not acting as a force. (Technically, if the body is larger than a point, it can have tidal forces acting upon it, which are forces that occur because of a differential in the gravitational effect between the two ends of the body, but we'll ignore those.)
 
Ok, so let's look a little deeper into these geodesic things. What do they look like? Standing on the surface of the Earth, if we throw a ball into the air, it will trace out a parabola through space as it rises and then falls back down to Earth. This is the geodesic that it follows. It turns out that given the appropriate definition, this path is the equivalent of a straight line through four-dimensional spacetime, given the bending of spacetime. How does this relate to what we think of as the acceleration due to gravity?
 
Let us choose a coordinate system based on our location on the Earth. We'll say that I'm at the origin, and define that we throw the ball up in the air at time t = 0 (this is essentially giving a name to the location, nothing more). We can describe the position of the ball in spacetime in this coordinate system using an appropriate parameter (that we call an "affine parameter"). As the ball moves through spacetime, its position in spacetime is given by appropriate functions of this parameter. We can rewrite things slightly, to relate its position in space to its position in time. Then, when we look at this trajectory, it appears that the object is accelerating towards the earth, giving rise to the idea that gravity is acting as a force.
What is really happening, however, is that the object's motion in our coordinate system is described by the geodesic equation. If you want some math, this equation looks like the following:

Here, x (with superscript Greek indices) describes the position of the ball in our coordinate system. The indices indicate whether we're talking about the x,y,z or time coordinate. The parameter t that the derivatives are being taken with respect to is the affine parameter; in this case, it is known as the "proper time" of the object (for slowly moving objects, we can think of t as the time coordinate in our coordinate system). The first term in this equation is the acceleration of the object in our coordinate system. The second term describes the effect of gravity. The thing that looks like part of a hangman's game is called a connection symbol. It encodes all of the effects of the bending of space time (as well as information about our choice of coordinate system). There are actually sixteen terms here: it's written in a convention called Einstein summation convention. This shows that the effects of the bending of spacetime change the acceleration of an object, based on its velocity through not only space but also through time.
 
If there is no curvature to spacetime, then the connection symbols are all zero, and we see that an object moves with zero acceleration (constant velocity) unless acted upon by an external force (which would replace the zero on the right-hand side of this equation). (Again, there are some technicalities: this is only true in a Cartesian coordinate system; in something like polar coordinates, the connection symbols may not be vanishing, but they're just describing the vagaries of the coordinate system in that case.)
 
If there is some bending to spacetime, then the connection symbols are not zero, and all of a sudden, there appears to be an acceleration. It is this curvature of spacetime that gives rise to what we interpret as gravitational acceleration. Note that there is no mass in this equation - it doesn't matter what the mass of the object is, they all follow the same geodesic (so long as it's not mass-less, in which case things are a little different). So, what good is this geodesic description of the force of gravity? Can't we just think of gravity as a force and be done with it?
 
It turns out that there are two cases where this description of the effect of gravity gives vastly different results compared to the concept of gravity as a force. The first is for objects moving very very fast, close to the speed of light. Newtonian gravity doesn't correctly account for the effect of the energy of the object in this case. A particularly important example is for exactly mass-less particles, such as photons (light). One of the first experimental confirmations of general relativity was that light can be deflected by a mass, such as the sun. Another effect related to light is that as light travels up through the earth's gravitational field, it loses energy. This was actually predicted before general relativity, by considering conservation of energy with a radioactive particle in the earth's gravitational field. However, although the effect was discovered, it had no description in terms of Newtonian gravity.
 
The second case in which the effect of gravity vastly differs is in the realm of extremely strong gravitational fields, such as those around black holes. Here, the effect of gravity is so severe that not even light can escape from the gravitational pull of such an object. Again, this effect was calculated in Newtonian gravity by thinking about the escape velocity of a body, and contemplating what happens when it gets larger than the speed of light. Surprisingly, the answer you arrive at is exactly the same as in general relativity. However, as light is mass-less, you once again do not have a good description of this effect in terms of Newtonian gravity, which tells you that there has to be a more complete theory.
So, to summarize this part, general relativity says that matter bends spacetime, and the effect of that bending of spacetime is to create a generalized kind of force that acts on objects. However, it isn't a force as such that acts on the object, but rather just the object following its geodesic path through spacetime.

 
HOW I KNOW GRAVITY IS NOT (JUST) A FORCE

 
When we think of gravity, we typically think of it as a force between masses. When you step on a scale, for example, the number on the scale represents the pull of the Earth’s gravity on your mass, giving you weight. It is easy to imagine the gravitational force of the Sun holding the planets in their orbits, or the gravitational pull of a black hole. Forces are easy to understand as pushes and pulls.
But we now understand that gravity as a force is only part of a more complex phenomenon described the theory of general relativity. While general relativity is an elegant theory, it’s a radical departure from the idea of gravity as a force. As Carl Sagan once said, “Extraordinary claims require extraordinary evidence,” and Einstein’s theory is a very extraordinary claim. But it turns out there are several extraordinary experiments that confirm the curvature of space and time.
The key to general relativity lies in the fact that everything in a gravitational field falls at the same rate. Stand on the Moon and drop a hammer and a feather, and they will hit the surface at the same time. The same is true for any object regardless of its mass or physical makeup, and this is known as the equivalence principle.
Since everything falls in the same way regardless of its mass, it means that without some external point of reference, a free-floating observer far from gravitational sources and a free-falling observer in the gravitational field of a massive body each have the same experience. For example, astronauts in the space station look as if they are floating without gravity. Actually, the gravitational pull of the Earth on the space station is nearly as strong as it is at the surface. The difference is that the space station (and everything in it) is falling. The space station is in orbit, which means it is literally falling around the Earth.
 
This equivalence between floating and falling is what Einstein used to develop his theory. In general relativity, gravity is not a force between masses. Instead gravity is an effect of the warping of space and time in the presence of mass. Without a force acting upon it, an object will move in a straight line. If you draw a line on a sheet of paper, and then twist or bend the paper, the line will no longer appear straight. In the same way, the straight path of an object is bent when space and time is bent. This explains why all objects fall at the same rate. The gravity warps spacetime in a particular way, so the straight paths of all objects are bent in the same way near the Earth.
So what kind of experiment could possibly prove that gravity is warped spacetime? One stems from the fact that light can be deflected by a nearby mass. It is often argued that since light has no mass, it shouldn’t be deflected by the gravitational force of a body. This isn’t quite correct. Since light has energy, and by special relativity mass and energy are equivalent, Newton’s gravitational theory predicts that light would be deflected slightly by a nearby mass. The difference is that general relativity predicts it will be deflected twice as much.
The effect was first observed by Arthur Eddington in 1919. Eddington traveled to the island of Principe off the coast of West Africa to photograph a total eclipse. He had taken photos of the same region of the sky sometime earlier. By comparing the eclipse photos and the earlier photos of the same sky, Eddington was able to show the apparent position of stars shifted when the Sun was near. The amount of deflection agreed with Einstein, and not Newton. Since then we’ve seen a similar effect where the light of distant quasars and galaxies are deflected by closer masses. It is often referred to as gravitational lensing, and it has been used to measure the masses of galaxies, and even see the effects of dark matter.
Another piece of evidence is known as the time-delay experiment. The mass of the Sun warps space near it, therefore light passing near the Sun is doesn’t travel in a perfectly straight line. Instead it travels along a slightly curved path that is a bit longer. This means light from a planet on the other side of the solar system from Earth reaches us a tiny bit later than we would otherwise expect. The first measurement of this time delay was in the late 1960s by Irwin Shapiro. Radio signals were bounced off Venus from Earth when the two planets were almost on opposite sides of the sun. The measured delay of the signals’ round trip was about 200 microseconds, just as predicted by general relativity. This effect is now known as the Shapiro time delay, and it means the average speed of light (as determined by the travel time) is slightly slower than the (always constant) instantaneous speed of light.
When I performed a research analysis on the Earth’s shape and its Gravity field, I was truly indicted to know that the gravitational potential of a perfectly uniform sphere would be equal at all points on its surface. However, the Earth is not a perfect sphere; it is oblate spheroid, and has a smaller radius at the poles than at the equator. Surveys in the early eighteenth century, under the direction of Ch-M de La Condamine and M de Maupertius found that a meridian degree measured at Quito, Equador, near the equator, was about 1500 m longer than a meridian degree near Tornio, Finland, near the Arctic circle. Subsequently, various standard reference spheroids or ellipsoids have been proposed as first-order approximations to the shape of the Earth, such as the World Geodetic System 1984 . Given such an ellipsoid, a gravity field can be calculated analytically as a function of latitude. For example, a reference gravity formula was adopted by the International Association of Geodesy in 1967 (IGF67, Table 1), and another introduced in 1984 (WGS84, Table 1).
The mean density of the Earth, which is fundamental to the calculation of gravitational attraction, was first estimated following an experiment in 1775 by the Rev. Neville Maskelyne, using a technique suggested by Newton. If the Earth was perfectly spherical and of uniform density, then a plumbline would point down towards the centre of the Earth because of the force of gravity on the bob. However, any nearby mass would deflect the plumbline off this ‘vertical’. Maskelyne and his co-workers measured plumb-bob deflections on the Scottish mountain, Schiehallion (Figure 1). They discovered that the mountain's gravitational pull deflected the plumb line by 11.7 seconds of arc. This allowed Charles Hutton to report in 1778 that the mean density of the Earth was approximately 4500 kg m−3. This density value leads to an estimate of the mass of the Earth of about 5 × 1024 kg, not far from the currently accepted value of 5.97 × 1024 kg. The Schiehallion experiment had another distinction, in that in order to calculate the mass and centre of gravity of the mountain a detailed survey was carried out, and the contour map was invented by Hutton to present the data. The link to the image is: Data Map.
Since the mass of the Earth is not distributed uniformly, the real gravity field does not correspond to that calculated for an ellipsoid of uniform density. The ‘geoid’ is a surface which is defined by points of equal gravitational potential or equipotential, which is chosen to coincide, on average, with mean sea-level. The geoid is not a perfect ellipsoid, because local and regional mass anomalies perturb the gravitational potential surface in their vicinity by several tens of metres. For example, a seamount on the ocean floor, which is denser than the surrounding seawater, will deflect the geoid downwards above it. ‘Geoid anomalies’ are defined as displacements of the geoid above or below a selected ellipsoid. The concept of the geoid as the global mean sea-level surface can be extended across areas occupied by land. This provides both a horizontal reference datum and a definition of the direction of the vertical, as a plumbline will hang perpendicular to the geoid. To this point, I still wonder what gravity actually is.

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