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Using the Newtonian EP, one can replace F = GMm/r2 by a = GM/r .
This embodies the EP since the value of a clearly does not depend on the object affected. A such, gravity can be considered as "an acceleration field on the medium".
Calling r(t) of a free fall particle its "worldline", we can see that since a = r..(t), it is the curvature of the free fall particle's worldline, and so we can summarize Newtonian gravity as:
"a mass M sets up a worldline-curvature field on the medium about it according to:
worldline curvature = GM/r2 "
Clearly the worldline-curvature of all objects in that region are affected accordingly.
Since r(t) is a line in a time-space graph, we consider it informally as a "time-space" quantity, and the worldline-curvature field is a field in this (t,r) "time-space".
According to the EP a free fall particle is locally-inertial, and taking this fully seriously as Einstein did, leads us to conclude that a free fall particle has a geodesic worldline, which implies that the spacetime must be curved.
Also, taking seriously the fact that inertial forces are indistinguishable from gravitational forces, one concludes that the Earth-stationary observer is accelerated, and so its worldline is necessarily curved. Consequently, the free fall particle has a geodesic worldline and the stationary observer watching it 'fall' has a curved worldine, the opposite of what is reported by Newtonian gravity theory.
This "switched perspective" is based on Einstein's understanding that if phenomena are indistinguishable, they are identical, thus leading him to say - based on the EP - that free fall is not just locally indistinguishable from inertial, it IS inertial.
It is clear that the relative acceleration measured between the free fall particle and the Earth-stationary observer is the same whether one works within the Newtonian framework or that of the curved spacetime theory, and so it is clear that in the Newtonian regime the curvature ascribed to the stationary observer's worldline in the curved spacetime theory must equal numerically the curvature ascribed to the free fall particle's worldline in Newtonian theory.
So the way to get from Newtonian gravity theory to the curved spacetime theory is to "switch perspectives", ascribing the curvature not to the worldline of the particle, ie not to a field-value on the Newtonian 'medium', but rather relating it to a curvature-related property of the spacetime (more precisely, it is related to "the connection"; see more about this below).
Geodesics are by definition straight, but if we were so sloppy as to say that around a mass the geodesics of spacetime are curved, and that explains why inertial particles are claimed by Newtonian theory to accelerate in the presence of mass, then we would also say that the field of worldline-curvature which Newtonian theory presents is actually - in the 'switched perspective' - a field of the geodesic's curvature. But of course geodesics are not curved.
(To get the idea straight one must understand that the tidal effect in Newtonian theory is evidence of non-zero Riemann curvature on spacetime, and to understand how Riemann curvature arises from a connection; see more about this below.)
The falling of the apple as Synge would have it:
Calculation in Synge, below: Since a free fall frame does not experience inertial forces, it IS inertial, and as with any inertial frame it can consider itself not moving at all, and so it has no change in spatial coordinates, only time passes, so dr/ds = 0. Below, the coordinate 1 is r.
Any non-geodesic worldline has curvature, and so the Earth-stationary observer seeing the apple coming closer experiences inertial forces, is accelerated, and their worldline has curvature. The amount of curvature is given by the geodesic equation, which says for a geodesic that the curvature is zero: schematically: 0 = d2x/ds2 + Gamma . For the Earth-stationary observer the curvature or acceleration is NOT zero, but since it is stationary its r-component as calculated in its frame is zero: most of the components of the acceleration vector ARE zero, but the r-component of its acceleration can be calculated as follows:
ar = d2r/ds2 + Gammartt dr/ds dr/ds = 0 + Gammartt dt/ds dt/ds. In the Newtonian regime dt/ds is about 1. So since there is this extra piece, the time-component dt/ds not just dr/ds, d theta/ds etc, there is a non-zero "acceleration" of the Earth-stationary observer!
See calculation below:
if something experiences inertial-forces, as does the Earth stationary frame, it is accelerating and so its worldline is curved. That's a basic fact. And one can calculate that worldline-curvature, ie acceleration, as Synge did, see the calculation there.
How it comes about mathematically that there is acceleration: acceleration is a 4-vector not a 3-vector as in Newtonian gravity. It has a t-component. And there is mixing of space and time, not just space and time attached to each other in the ordinary separate "rectangular" way of classical physics. So the r-component of the 4-acceleration as in Synge's calculation is affected by the existence of the t-dimension's contribution.
How it comes about physically that there is acceleration: the phenomena we call "Newtonian gravity" which arise from what can be represented as a warping of the t-axis, and this is something akin to a 'curving' of the t-line (but it NOT actually 'curving'), and thus there is by definition non-zero acceleration in the Earth-observer's frame.
For more explanation of all this, see the videos of my "Introduction to General Relativity" lectures 1 & 2 (alternatively, here for the unedited version), and see also lecture 7 there (including the Synge diagrams in videos #73 & 80).
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Some geometry
We are used to the fact that two neighboring straight lines cross each other only once, and if they are parallel it means they never meet. However we are also familiar with curved surfaces on which lines which might seem straight can cross each other in different parts of the curved surface, and lines which seem parallel at one place on the curved surface diverge or converge elsewhere. In mathematical terms we'd say these straight lines 'deviate' Of course this 'deviation'-behavior is a trivially-obvious consequence of the fact that the surface is curved.
What is not trivial is the fact that mathematically, it turns out that a line that is straight 'locally' on a curved surface has in fact every single relevant property of a straight line on a flat surface. Astonishingly, there is no way to locally distinguish one from the other! And in every region of the curved surface, when we look locally, it has every characteristic of straightness. And so the ONLY way we can know whether a locally-straight line is on a flat surface or on a curved surface is to extend them so that they go beyond the small local region, and then we can see whether they behave in ways that indicate curvature of the surface, ie whether they 'deviate'.
Note that they have every straightness-property locally - but 'globally' in the sense that this is true locally wherever we are on the surface. This means that they are really straight. In math we call such lines 'geodesics'.
On a curved surface of course these geodesic deviate, and indeed geodesic deviation is the hallmark of curvature.
Another astonishing fact is that one can define mathematical spaces which have the same geometrical properties, they have straightness defined on them, and these straight lines can deviate. By analogy to our experience with curved surfaces we call such a space 'curved'.
Terminology: The geometric property which grants a manifold the ability to have straight lines (or which describes this ability) is called the 'connection'. Mathematically we say that a space with a connection has geodesics. Or that if a space has geodesics, obviously it must have a connection.
When there is geodesic deviation, we say there is "non-zero Riemann curvature". On a flat surface where there is no geodesic deviation, there is zero Riemann curvature. That is, in any space which has a connection we can define "Riemann curvature", measured by geodesic deviation (and also other ways) just that in a flat space the amount measured is zero.
Back to physics
Here's an example of what we just discussed above: an inertial particle's r(t) is a straight line in (t,r). We can say that since "straightness" is defined on it, this "time-space" has the geometric property of a "connection", and inertial worldlines are geodesics of 2-d "time-space". (Actually of course we know that spacetime is 4-d, and so we understand that inertial worldlines are geodesic of this 4-d spacetime.)
As soon as we consider 2-d time-space (or 4-d spacetime) to have this sophisticated geometric property, and we know about curvature of spaces, we may wonder whether it is possible that physical time-space can also be curved! What would it mean - what would we observe physically if this were true (ie if there is non-zero Riemann curvature)? The answer of course is deviation. We would need to see two neighboring geodesics behaving like straight lines locally everywhere, but nevertheless deviating.
In the case of time-space, this means we would see two particles which are inertial everywhere locally, but nevertheless act non-inertially overall.
We can see that the prosaic gravitational tidal effect so well-known in Newtonian theory but as interpreted within the perspective of Einstein's EP indicates exactly that. Neighboring free-fall particles are each inertial, and will be so throughout their 'fall', yet it is clear that there is relative acceleraiton between them! From the mathematical perspective, this means that neighboring geodesics are deviating, ie the physical phenomenon of "relative acceleration of neighboring inertial particles" ("RANIP") correlates exactly to the mathematical property we call "geodesic deviation". in analogy with the situation of curved surfaces, where we observe geodesic deviation, we can use the term 'curved' to describe any space in which there is geodesic deviation.
Thus as a result of the tidal effect as interpreted by Einstein, and in analogy with the situation of curved surfaces, where we observe geodesic deviation, we can say that spacetime is "curved", ie it has non-zero Riemann curvature.
Now we can re-visit our earlier statement: we said "Geodesics are by definition straight, but if we were so sloppy as to say that around a mass the geodesics of spacetime are curved, and that explains why inertial particles are claimed by Newtonian theory to accelerate in the presence of mass, then we would also say that the field of worldline-curvature which Newtonian theory presents is actually - in the 'switched perspective' - a field of the geodesic's curvature. But of course geodesics are not curved."
Now we can make a more sensible statement: Free particles, in other worlds far away from any masses, have worldlines which are geodesics. Geodesics are STRAIGHT. But they can deviate if the space is curved. Around a mass, spacetime is curved, and so although free particles in that region remain free, and their worldlines are still geodesic, however as a necessary result of the curvature of the space of course neighboring geodesics deviate.
From the perspective of Newtonian theory, free fall particles have curved wordlines, but really these free fall worldlines are geodesics, just that they are geodesics of a curved space.
Note that the curvature of the space is measured by the amount of the geodesic deviation - not by the 'curvature of the geodesics', since geodesics are NOT curved. Obviously though, there is a close relation between the amount of worldline-curvature of a free fall particle as seen in Newtonian theory and the amount of Riemann-curvature of the spacetime. However the former is a property of an individual particle whereas the latter is a relative measure involving two neighboring free fall particles. It is not surprising that the latter, the Riemann curvature, is therefore something like the derivative of the former, which is a measure of the connection, as the tidal acceleration is something like the derivative of the gravitational acceleration.
............Newton-Cartan theory...........
Newton-Cartan theory is a formal framework presenting this idea of gravity being a (non-trivial) connection on spacetime, giving rise to non-zero Riemann curvature. That one must only 'switch perspectives' to get from EP-based Newtonian gravity to Newton-Cartan theory indicates how similar they are, and perhaps explains why some results of Newtonian theory can easily be adapted to become results of a curved spacetime theory of gravity, such as Einstein's general relativity.
See for example these lecture-videos:Newtonian gravity as a flatspacetime 'shadow' of Einstein's curved spacetime general relativity & Magically obtaining Einstein from Newton, via a reverse Platonic projection.
Einstein's theory differs from Newton-Cartan theory mostly in that an additional geometric structure is imposed on spacetime - a metric.
That is, besides the connection and the Riemann curvature arising from it, there is an additional geometric structure. In other words, besides being able to define straightness and curvature in spacetime, there is an additional property to spacetime which can be measured - 'spacetime distance'. Newton-Cartan produces essentially the phenomena of Newtonian gravity, but as a result of it being a theory of a metric (which gives rise to a connection which gives rise to Riemann curvature) GR is far richer geometrically, and produces more physical phenomena.
In ordinary life we can see that there is obviously a metric on space, and a metric on time, but we certainly have no direct intuition of 'spacetime', let alone a metric on it. Newton-Cartan theory adds the notion of spacetime, and of a connection on it: the trivial connection is supplied by inertia, since inertial r(t)'s are straight, ie inertial worldlines are geodesics of spacetime, and the non-trivial connection is supplied by gravity. However, there is no spacetime metric even in Newton-Cartan theory - Einstein's general relativity however adds a spacetime metric, and that provides its essential advantage (see excerpts from MTW below).
After seeing the videos of my "Introduction to General Relativity" course, you are welcome to read the accompanying text-book (the lengthy Preface/Introduction is here), in which there are not only many explicit references to excellent texts such as those of Synge and MTW, but also guides to "must-readings" in these magnificent works which you would be able to follow after completing the videos and the accompanying textbook.
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Answers to questions posed by viewers of the first few Youtube-videos
Regarding the relative acceleration experienced by two particles in a free falling 'elevator': Of course Newtonian gravity also states that they will experience relative acceleration, called the "gravitational tidal acceleration", but in that theory the two particles are NOT considered to be inertial, they are after all being "pulled by gravity", which is a force, and so they accelerate. In GR, gravity is NOT a 'force', and neither of the two particles are "being pulled" and they are NOT accelerated, they are inertia.
Of course GR agrees on the observable phenomenon, the relative acceleration (which Newtonian gravity calls "tidal acceleration") just that in GR the two particles which have this indisputable relative acceleration are both inertial! This would be impossible if the universe were as the Newtonian conception had it, but one can show that it is exactly the effect we would expect from a warping of time (as demonstrated heuristically in various of the videos).
Note re the "time-warping" I mention in the lectures: Actually, GR tells us that spacetime is warped, not just time itself, and as a result there are various phenomena beyond those predicted by Newtonian gravity, for example spatial curvature, and for a spinning mass like our planet there is also 'frame-dragging' (you can about it on the web). Just that in order to produce only those phenomena which were part of Newtonian gravity, one would only need to warp time alone, not space.
Of course in order for a theory to be 'covariant' (which Einstein at first did not believe was a 'must' and then changed his mind, and is now considered by physics as sacrosanct, but read also about Kretschman's clarification etc), we cannot have just time warping, since we cannot separate time of from spacetime in that arbitrary way.
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