from my book

Contrary to popular misconceptions, although the local laws of classical physics have the same “tensor” and/or “spinor” form for all motions of detectors measuring all the observable possessed by the “test particles,” nevertheless, there still are privileged geodesic force-free dynamical motions of the test particles in Einstein’s two theories of relativity special 1905 and general 1916.[i]  This was in Einstein’s words “My happiest thought.”

“The breakthrough came suddenly one day. I was sitting on a chair in my patent office in Bern. Suddenly the thought struck me: If a man falls freely, he would not feel his own weight. I was taken aback. This simple thought experiment made a deep impression on me. This led me to the theory of gravity. I continued my thought: A falling man is accelerated. Then what he feels and judges is happening in the accelerated frame of reference. I decided to extend the theory of relativity to the reference frame with acceleration. I felt that in doing so I could solve the problem of gravity at the same time. A falling man does not feel his weight because in his reference frame there is a new gravitational field, which cancels the gravitational field due to the Earth. In the accelerated frame of reference, we need a new gravitational field.”

First note the date 1907. Einstein is using Newton's 1686 theory of gravity not his then future 1916 general relativity way of thinking that he has not yet created. Einstein is struggling with the wrong notion of “acceleration.”

"A falling man is accelerated."

Yes, in Newton, but not in Einstein nine years in the future! The falling man's frame is LIF with zero proper acceleration. In fact it's the surface of static LNIF Earth with proper radial acceleration upward rushing toward the falling man.

Proper acceleration of falling man = Relative 1905 SR kinematic acceleration - Proper acceleration of Earth

Proper acceleration of falling man = D2X/ds2

Relative 1905 SR kinematic acceleration = d2X/ds2

Proper acceleration of Earth = {STATIC LNIF EARTH}(dX/ds)(dX/ds)

X = relative separation test particle to detector on Earth.

{  } = Christoffel symbol used in the Levi-Civita connection

v = dX/ds

In fact when v/c << 1, the 3-vector piece of the above 4-vector equation is:

{STATIC LNIF EARTH}(dX/ds)(dX/ds) ~ -GMEarthr/r3

Proper acceleration of falling man = 0 because an accelerometer pinned to the man shows zero on its pointer. Therefore,

Relative kinematic acceleration = Proper acceleration of Earth

Where a Doppler radar measures the relative kinematic acceleration between the falling man and Earth. In contrast, a second accelerometer clamped to the detector at rest on surface of the Earth measures -GMEarthr/ras the weight divided by the mass of the detector.

“A falling man does not feel his weight because in his reference frame there is a new gravitational field, which cancels the gravitational field due to the Earth. In the accelerated frame of reference, we need a new gravitational field.”

That statement by Einstein in 1907 is how Newton would explain it. Einstein put himself in Newton's shoes for a moment. It's not the way his later 1916 matured GR explains it.

0 = Relative 1905 SR kinematic acceleration - Proper acceleration of Earth

This “cancellation”, the “0” on the above word equation is not a cancellation of two real dynamical fields. Einstein's unfortunate informal language in 1907 has no relevance to his, then, future theory.

"In the accelerated frame of reference, we need a new gravitational field."

That's the LIF, which is not accelerated in the sense of 1916 Einstein GR, but is accelerated in the different sense of 1686 Newton. These subtle oft unnoticed paradigm shifts in the meanings of “acceleration,” “inertia,” “inertial frame” cause many people a great deal of confusion even today,

Einstein was still muddled in 1907 as he struggled to make the great breakthrough. Your understanding is trite and superficial based on semantics and exaggeration of an early remark of Einstein's.

[i]  This geodesic premise is Newton’s first law of motion most generally expressed.

[ii] On the Relativity Principle and the Conclusions Drawn from It, Albert Einstein,

Jahrbuch der Radioaktivitat und Electronik 4 (1907) – Re-Published in three parts.

Am. J. Phys. 45, Part I - (6), June 1977, pp. 512-517; Part II – (9), September

1977, pp. 811-816, Part III - (Gravitational Part) – (10), October 1977, pp. 899-

902. This paper addresses only Part III – from Peter Brown’s paper.

In this EARLY 1907 quote Einstein (who is still under Newton’s magick without magic spell) means Newton's "accelerated frame", that is, dV(test particle)/ds in Newton's first law (geodesic equation) as written in modern POST-1907 GR language. Suppressing indices:

DV(test particle)/ds = dV(test particle)/ds - {LNIF detector}V2(test particle) = 0

The "cancellation" is precisely

dV(test particle)/ds - {LNIF detector}V2(test particle) = 0

In other words, in the general case that even applies to Newton's 2nd and 3rd laws is:

Einstein's proper tensor acceleration = Newton's apparent acceleration - fictitious LNIF inertial pseudo fictitious forces per unit test particle rest mass = real applied force to the test particle per unit test particle mass

Fictitious forces on test particle = Real forces on LNIF detector of test particle's motion

In the case of Newton's 3rd law, when Alice and Bob form an isolated closed system

DP(Alice + Bob)/ds = DP(Alice)/ds + DP(Bob)/ds = 0

Both must be measured in the same frame by Eve, i.e.,

DP(Alice or Bob)/ds = dP(Alice or Bob)/ds + {Eve}V(Alice or Bob)P

“I continued my thought: A falling man is accelerated. Gravity and inertia are interrelated." Einstein

Here is the source of the confusion.

Einstein is naturally thinking in Newtonian terms.

However, in GR terms that he still had not invented back then in 1907: "acceleration" above means relative kinematical acceleration between test particle and local frame. It does not mean real (proper) acceleration (off-geodesic) as measured by an accelerometer.

The general law is:

Real acceleration on test particle = relative kinematical acceleration between test particle and local frame - real acceleration of local frame.

DP(test particle)/ds = dP(test particle-frame)/ds - DP'(local frame)/ds

P = mV   for the test particle under observation by the local frame detector

V = dX/ds

X = relative kinematical displacement between test particle and local frame detector as measured by a Doppler radar clamped to the local frame.

D/ds = d/ds - {LC frame connection}dX/ds

DP(test particle)/ds

= dP(test particle)/ds - {LC frame connection}(dX/ds)P(test particle)

When dm/ds = 0, it follows that

D2X/ds2 = d2X/ds- {LC frame connection}(dX/ds) 2

{LC frame connection}(dX/ds) 2 = M-1DP(frame)/ds

M = mass of frame/detector

{LC frame connection} has dimension 1/Length

ds is the PROPER TIME element along world line of object.

Each term has an independent measurement technique.

Real accelerations are measured by accelerometers attached to the objects.

http://en.wikipedia.org/wiki/Accelerometer

Accelerometers measure off-geodesic "pushes" by real forces.

Doppler radars measure the kinematic acceleration.

Therefore,

DV/ds is measured directly locally by an accelerometer clamped to the test particle  - real measurement 1

dV/ds = d2X/dsis measured indirectly by the Doppler radar clamped to the local frame detector - real measurement 2

M-1DP(frame)/ds is measured directly by a second accelerometer clamped to the frame-Doppler radar - measurement 3

The BASIC LAW is

Measurement 1 = measurement 2 - measurement 3

Provided that test particle and frame Doppler radar are not far away from each other relative to the smallest local radius of curvature A1/2. The curvature is of order A-1

The geodesic equation is simply Newton's first law when

Measurement 1 = 0

Newton's second law is simply when

Measurement 1 =/= 0

There is never any cancellation of real forces on any one object in this context

The LNIF ---> LIF in measurement 3 simply means removing a real unbalanced force on the frame detector according to Newton's 1st law.

“Then what he feels and judges is happening in the accelerated frame of reference.” Einstein

Einstein's use of "accelerated" here is in Newton's sense - the rest frame of the freely falling man is kinematically accelerated relative to the Earth

i.e. d2X/ds2

The freely falling man's local frame is LIF - though Einstein did not yet discover that in 1907 and his informal language is still Newtonian because the modern GR informal language of 1916 and after is not yet emerged.

“There is a new gravitational field, which cancels the gravitational field due to the Earth.”  Einstein

This is Einstein's remark that physics cranks pull out of proper context. Yes, Einstein wrote it back around 1907 before he understood the problem the way he eventually would in 1916 and later.

In fact there is only one gravity field not two.

The point is that there was never a real gravity force field on the test particle to begin with.

Therefore, you don't need a second gravity force field to cancel what was never there!

Indeed, there is no way to measure either of these alleged two real gravity forces to begin with. You can never separate them. Accelerometers on test particles always show zero.

Therefore, like the Maxwellian 19th century mechanical aether that acts without being reacted upon that Einstein eliminated in 1905, these two ghostly independently unobservable-in-principle forces are not independently measurable - they are errors of thinking - excess metaphysical informal language baggage. Even the great Einstein got muddled temporarily on this one, though with good reason. Unfortunately many people today who should know better remain muddled. If gravity is not a real force like the electro-weak-strong forces, then what does it mean to unify them?