When General Relativity (GR) is mentioned, what comes to mind is the picture of a ball sitting on an 'hammock' and 'weighing it down'. It is said that a massive object warps space time in the vicinity of a massive body. Notice that we are not told that the body warps the space time only at the region the object is sitting on. On the contrary, how the space time curves at where the moon is is supposed to be affected by the earth. So you expect that the equations of GR should relate the curvature of space time to the distant object at distance r, just like Newton's law of gravity relates force to the distant mass. However, you are totally mistaken! GR is a miserable theory that fails even to relate effect to its alleged cause!
Newton's law of gravity can verbally be stated as that a massive object attracts another massive object at a distance with a force that diminishes with the distance. This 'verbal' statement agrees perfectly with its mathematical statement. So Newton used the mathematics only to state the quantities therein in a more precise way. GR on the other hand uses arcane mathematics to effectively 'swap the hat and pull out the rabbit', a shell game very prevalent in modern theoretical physics! The notion than in physics, we talk verbally and then spit out some equation allows the theoreticians to get away with explaining a single phenomenon using two different or even contradictory theories while insisting that 'it is one throry'! The words speaks one thing, the equations speaks another thing!
In Newton's law of gravity, it is quite straight foward and the equation goes:
F=GmM/r^2
F=force
G=gravitational constant
m=the smaller mass
M=the bigger mass
r=distance between m and M
So the statement is very clear; an object of mass m, at distance r away from another object of mass M experiences a force of F that tries to pull it towards the object of mass M. So the verbal statement that 'it is the object of mass M that pulls the other object of mass m over distance is just restated by the equation albeit giving us the exact way the quantities are related. I say that 'the equation does relates the cause (object of mass M), to its alleged effect (the force F on the object of mass m).' The Newton's law does fulfils the very first task that a physics theory should be expected to fulfil. We just cannot have a verbal statement that 'the force F is caused by object M', while the equation is talking something else.
Now lets come back to GR. They say that the massive object, (eg the sun), causes the space time to curve in the vicinity of the massive object. So we expect the equation of GR to tell us something like 'given an object of mass M the spacetime is curved by such amount at distance r from the object'. So if you were to just take a glance at the GR equation, you should expect to see 'r' and 'M' somewhere etc. However, you don't see anything like that! At the left, we have a differential equation and at the right, we have energy density. Actually, you will find that the GR equation tells us that 'the energy density at point P, curves the spacetime at the same point, P, by such and such amount'. It tells us nothing about how the energy density curves the spacetime at any other place! However, this 'energy (or mass) curves spacetime at some distance away' is all that they verbally state as what GR says! This is why you are shown an object 'weighing down the hammock', ie an object curving the 'hammock' even at the region away from where the object is sitting on. Then of course they insist that it is the mathematical statement that 'accurately' presents the GR theory!
Now to be sure you get it very well, there is a way of stating Newton's law of gravity in terms of how a massive object causes 'acceleration' right at the region where the object is sitting in, and apparently, it tells us nothing about how it affects acceleration anywhere! This is called 'poison's equation'. So we might say that the GR equation, i.e. the Einstein's Field Equation' (EFE), is an analogy of poison's equation. But there is a trick that Einstein overlooked. The poison's equation (PE) comes from the following statement: 'the total (gravitational) flux crossing any closed surface is proportional to the mass enclosed by that surface'. This is called 'Gauss's Law', and is one way of stating the Newton's law. PE states that the 'convergence' of flux to a point is proportional to the (infinitesimal) mass density sitting at that point. So in this way of relating a quantity to a mass in situ, PE is analogous to EFE. However, how flux converges to a point is of course related to how it crosses some surface enclosing that point as they head to the point of convergence. It is clear that the flux will not converge to an 'empty space' but it will just past through that point. So equating the convergence to zero says something CORRECT about the behaviour of the flux at that 'empty' point, that the flux is not converging there, but it fails to tell us something else that is crucial: there is still a flux that is on its way to converging somewhere else. This is stated by the Gauss's Law. EFE fails to tell us this because it is only an analogy of PE but GR doesn't give us a corresponding analogy of Gauss's Law.
So in completing the analogy, it is not enough for GR to tell us that the Ricci curvature in an 'empty' area is zero. It must also tells us that in general, there is another 'non-Ricci' curvature that is related to a distant mass. This is how PE works; with us having Gauss's Law at the back of the mind. By relating the 'no-Ricci curvature' that exists even in an 'empty space' to a distant mass, GR would have indeed related the effect to its alleged cause. But GR just stops at saying Rυν=0, and proceeds as though this tells us all there is to say about an 'empty space'! In effect, they give us an equation for 'empty space' that is indistinguishable from an equation that would describe a completely empty universe!
By simply saying that a point in space time is 'Ricci-flat', we will not be describing a specific empty location away from a given mass, e.g. some distance away from the sun, contrary to how GR will mis-apply the Ricci=0 equation. On the contrary, it describes every 'empty space', from an 'empty' place that is light years away from any matter to a completely empty universe! Consequently, the equation just tells us nothing but how something can wiggle or not without being Ricci curved. This is like telling us how something can curve along only one direction without curving along any other direction, e.g. how a cylinder curves as opposed to how a ball does so. So there are infact infinite ways in which something can curve but remaining Ricci 'flat'. All a GR advocate have to do is simply pick one of the infinite curvings allowed by the simple Ricci 'flatness' constraint that agrees with experiments at a given region and then claim that the Rυν=0 is in fact describing that specific region! It is an hilarious fat lie!
Comments
Now they claim that the choice of this constant to be m is to make the theory 'reproduce Newt's Theory when m is small'. But as you should see, setting m close to 0 does not yield anything like Newt's law of Gravity. It merely yield a Pythagoras theorem in a 'flat space'. This yields 2 problems with their theory:
1.)Newt's theory is not just an assertion that 'space time is flat'. It says how objects should behave in the vicinity of gravitating bodies
2.)They tell us that in GR, gravity is strictly a 'space time curvature' so as space-time tends to flatness, gravity should tend to cease existing, rather than tending to be Newtonian!
It turns out that the only equation that should have reproduced Newt's law for small m is the one that contain m, not a 'solution' for such 'solutions' only yield metrics and Newt's law doesn't talk about metrices! But as we have seen, EFE does not reproduce Newt's Law when we set m=0! Infact the resultant equation: Rμν=0 still describes a 'curved space'. So this further highlights the problem with GR that I have been trying to point out.
A theory that would truely reproduce Newt's theory is the one that relates accelerations to mass. This theory would modify the accelerations by a factor that depends on curvature at the point of acceleration. When the point is 'flat', then Newt's Law is reproduced. This curvature would, in turn, depends on the distant mass so that it tends to 'flatness' as m tends to zero.
http://ion.uwinnipeg.ca/~vincent/4500.6-001/Cosmology/Schwarzschild...
So in a similar way, GR should tell us that; 'though an empty space is Ricci Flat (Rμυ=0), the 'empty space' around the massive object is still curved in a different way (other than Ricci curvature) with a curvature that is related to the surrounded, massive object.' Thus 'Ricci curvature' is analogous to 'convergence of gravitational field' and the other curvature is analogous to 'gravitational field that is on its way to converging somewhere else'. (gravitational field lines that are 'on their way to converging somewhere else' bends towards each other but without meeting. So the 'convergence' here is 'meeting' and not just 'getting closer to each other'. in a similar way, there can be a 'curvature' without there being a 'Ricci Curvature').
But in 'curved space', such a theory that would reproduce Gauss's Law encounters insurmountable barrier. Noting that 'gravitational field' is just 'acceleration', a GR equation could have as well contain accelerations, rather than 'curvatures', albeit accelerations adjusted by the curvatures. So Einstein could have written a 'law' of gravity that contains accelerations that are modified by a factor that is related to the 'curvature of space time' at the point through which acceleration is happening'. When this curvature is zero, the theory should reproduce Newt's theory (preferably in Gauss's Law form).
But you can see that Einstein's Field Equation doesn't reproduce Newt's Law when mass is 'very small' for even if we set m=0, we just have Rμν=0 , which, as they agree, still described a 'curved space' such as in the 'empty' area around a massive object. So it is disingenuous for them to say that GR 'reproduces Newt's law for small mass'! Infact GR allows for anything around massive objects, even those that 'infinitely' contradicts Newt even 'for small masses'. They just artificialy pick one of the infinite so called 'solutions' allowed by GR, that agrees with experiments, and thus that agrees with Newt 'for small masses'! They lack an equation that would reproduce Newt's law in that it would not present an ambiguity of 'empty space around a massive object' when we set m=0. To do that, the 'm' in the equation must reference a distant object which is causing a 'curvature' at a distance. Then 'm=0' due to 'absence of mass in situ' is clearly differentiated from 'm=0' for 'smallness of the distant mass'. The latter should result in reproduction of Newt's law while the former should not necessarily do so as mass in situ can be zero but in an empty place around a gigantic matter.
Another problem is that relativist tells us that 'gravity is not a force' but a 'curvature of space time'. This is problematic because when we thus 'flatten the space time', (which should be by setting m=0), We have no gravity at all, yet we should rather reproduce the Newt's law! Then finally, in Gauss's Law, we have a global way of comparing vectors, eg accelerations. So we encounter no problem in the question of vectors that spans all the way round a Gaussian Surface. A curved space, on the other hand, forbids us from comparing distant vectors! Thus we can't talk of 'accelerations around a closed surface' that would reproduce Gauss's Law when the space time approaches flatness!
To be sure you get it well, even in Newt's gravity, there are infinite ways an object can move in the area around a massive object. But all these trajectories are constrained by the massive object that is exerting the gravitational force. They are constrained to accelerate towards the massive object according to the Newt's law. This constraint is specific in a specific location at a specific distance away from the specific massive object. But in GR, the 'curvature' is constrained just by the 'Rμυ=0', a condition that applies to every empty space, no matter what distance away from a massive object, or even in 100% empty universe! The 'Rμν=0' does not, in any way consider the mass configurations in the area around the 'empty space' that would supposedly create the specific curvature that will satisfy the 'Rμν=0'! Instead, there are countless configurations that satisfies 'Rμν=0'. GR just chooses one in an endless ocean of allowed configurations that satisfies measurements at a given region, and then insist that it is the GR, which has just said 'Rμν=0' without mensioning any mass, that 'predicts' that such a mass configuration results in the thus measured curvature! What nonsense!!