Consider the diagrams representing the duality between the weak and strong principles of equivalence. Now based on how these diagrams were taught to us (at least how they were taught to me); each of the two graphics are displayed for the sake of demonstrating how two separate situations are actually physically indistinguishable for the inside observer (Einstein himself).

enter image description here

After careful consideration and analyzation, I claim that these diagrams in no way admits any information which can be used to suggest that in each diagram the two scenarios are in any way physically or situation-ally nonequivalent. That is; free fall may not be described as any sort of new class of inertial motion as the Wikipedia article on "General Relativity" suggests. Equivalently gravitational forces may not be thought of as a new class of accelerated motion.

In my opinion this idea that free fall represents a new class of inertial motion is simply an artifact of the assumption that gravitational fields are not present at a distance far removed from a gravitational source. Essentially I believe all frames in which we don't feel inertial forces must be free falling frames (such as orbit or flat space drift) Even though it initially doesn't seem that way (Since the gravitational source cannot be seen from a far distance). Perhaps this is already considered obvious to some but I know it is not to everyone (Since so much work is being done to understand inertia by other means...such as say radiation reaction forces(1))

I don't want anyone to misinterpret my meaning; I COMPLETELY agree with the principle of equivalence, in fact I believe it is far more powerful than most physicists give it credit for. I only claim that the diagrams are misleading because each diagram uses two seemingly different yet already fully equivalent scenarios as a means to describe how they are actually (surprise!) physically equivalent from the point of view of the inside of an elevator but makes no claims about the equivalence on the outside as well; In each diagram there is an indirect suggestion being made; that the two situations are actually distinguishable. It is as if Einstein pops his head out the door and says "Oh I thought I was free falling, turns out my elevator was just drifting in locally Minkowskian space-time, silly me". I say he was always free falling, just not in the traditional way. In my opinion all Minkowskian frames are free falling ones.

The geodesic equation says that mass-energy follows its natural geodesic path through space-time but mass can only be measured via force probe if there is a resistance to that natural geodesic path. When you accelerate an object in a way which admits an inertial property of matter(One that is resisting its natural geodesic path through space-time) such as... the pushing force on our backsides as we accelerate in car, it MUST be the result of an objects coupling to the gravitational sources throughout the universe (and Earth too!) being resisted. This is essentially the meaning of a Machian principle of equivalence.

$\frac{d^{2}x^{\lambda }}{d\tau^{2} }+\Gamma ^{\lambda }_{\mu \nu }\frac{dx^{\mu }}{d\tau}\frac{dx^{\nu }}{d\tau }=0$

For the sake of clarity I will augment the geodesic equation to demonstrate how inertia may be calculated as an Machian effect of changing the natural geodesic path of an object. (The point here is to show that the connection coefficient is BUILT from all mass energy in the universe.)

$m\frac{d^{2}x^{\lambda }}{d\tau^{2} }+m\Gamma ^{\lambda }_{\mu \nu }\left ( \left ( T_{\mu \nu } \right )_{Universe} \right )\frac{dx^{\mu }}{d\tau}\frac{dx^{\nu }}{d\tau }=F^{\lambda }_{inertial}$

Also when most of us speak of the geodesic equation, we refer to it's description of an object in an inertial frame since it is always equated to zero but the equation is equivalent to the four force equation which is not necessarily zero. This is precisely why inertia manifests.

When an object is free falling.

$F^{\lambda }_{inertial}=0$

When an object is feeling inertial forces.

$F^{\lambda }_{inertial}\neq 0$

(1) I intend no disrespect to the legitimacy of radiation reaction forces or to those undertaking research in radiation reaction forces, only that if there are radiation reaction forces then they are available here, raining upon us on the surface of the Earth all the time since we are always standing in an accelerated frame. I would even go as far to say that radiation reaction research may serve as a means to understand inertia and gravitation far better than we currently do.


1a) Weight and inertia are one-in-the-same as told by the strong principle of equivalence.

1b) Applied forces and gravitational normal forces are one-in-the-same as told by the strong principle of equivalence.

2) Free fall, free drift and orbit are one in-the-same as told by the weak equivalence principle.

3) An objects mass-energy content can only be defined by effective measurement of an obstruction (by a force probe) to the object's natural geodesic path through space-time. For this reason mass-energy MUST be defined this way.

4) The principle of equivalence is Machian by virtue of the augmented geodesic equation. (NOT IN THE SENSE THAT MACH ORIGINALLY PROPOSED ~ SE USERS: PLEASE DO NOT REFERENCE THIS VERSION OF MACH'S PRINCIPLE WHEN COMMENTING OR ANSWERING BECAUSE IT IS NOT THE CONCERN HERE) Yet most physicist claim any sort of Machian principle is erroneous.

Additional note: When I approached my professor with this information; I originally suggested that if all objects with mass-energy in the universe (which share time-like separations) contributes to the natural geodesic paths of every other object in the universe. Then inertia can be described in terms of a Machian framework due to the principle of equivalence itself. He claimed that my argument was useless since that no one has been able to explain WHY mass energy contributes to space-time curvature. While I agree that the mechanism which somehow curves space-time due to the presence of mass energy is not well understood, I disagree with him that the information is pointless since these ideas must remain valid once we finally do figure out the mystery behind gravitational mechanism. The certainty of a Machian process for inertial emergence should not change once we understand gravitational mechanism. In my opinion, the understanding of gravitation in terms of a Machian principle of equivalence seems beneficial for solving new problems or setting new boundary conditions.


1) Where is my logic/thought process flawed?

2) How can a Machian principle of equivalence of this type be refuted? (Since it is explicitly described by the geodesic equation and the principle of equivalence.)

3) If an object's mass energy content can only be defined by a measure of an obstruction to its natural geodesic path through space-time, then how can two separate interactions such as electromagnetism and gravitation both generate by experiment the exact same values for an objects mass-energy if the two interactions are not fundamentally one in the same? (Let me say that the lack of measurable gravitational forces and the availability of measurable electromagnetic forces at microscopic scales compared to macroscopic scales where the opposite is true seems all too artificial.)

  • $\begingroup$ This is a really long post and I must say I don't understand half of where you're going with it. Some comments, though: (1) Spacetime is always locally Minkowskian; no one would ever claim otherwise, and the only experiments that could in principle detect curvature are those that take place over extended regions. (2) We don't know the mass of the Sun by pushing on it with a calibrated spring; we know the mass by observing the Earth's orbit (again, over an extended region). $\endgroup$ – user10851 May 22 '14 at 1:13
  • $\begingroup$ (3) In all this, you haven't given the definitions of "Machian" or "principle of equivalence" that you are working with. You admit that there are different definitions of the former, and there are at least three versions of the latter. In fact, the utter disarray of the Wikipedia article is a testament to how no two physicists agree on the wording of the equivalence principle, and why all science is done with the mathematical statement of all metrics being locally diagonalizable. $\endgroup$ – user10851 May 22 '14 at 1:19
  • $\begingroup$ I agree with your assumptions, logic and conclusions. With regard to the 3rd question, using a different theory, I obtain the same conclusion, that gravity and electromagnetism interactions are fundamentally the same! $\endgroup$ – Guill Jun 11 '14 at 7:43
  • $\begingroup$ @Guill Which theory are you using to obtain the same conclusion? I am curious since I have been trying to obtain more useful information from this idea. $\endgroup$ – spacetimeengineer Jun 11 '14 at 14:09
  • $\begingroup$ @ChrisWhite I certainly don't disagree that physicists use orbital equations to find the values of masses of other stars all the time, however if I were to ask you to measure the mass of the sun without first knowing the mass of the Earth or any other celestial body for that matter, what say you then? $\endgroup$ – spacetimeengineer Jun 11 '14 at 14:15

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