Mass of Many Different Forms and Origins

I have been led to the idea that there is in Newtonian mechanics only one form of mass, gravitational mass, which is derived from a gravitational potential as primordial, wit inertial mass = gravitational mass as a consequence of universality of free fall and conservation of energy. This is a clear picture, which can be understood by everybody, and which has shown to work excellent in practice. 

This is not the idea of modern physics, which plays with several different forms of mass:  

• gravitational mass
• inertial mass
• rest mass (relativistic)
• Higgs mass. 

with different origins:

• gravitation
• Higgs mechanism and strong force
• relativity theory
• Higgs mechanism.

No doubt this shows a very complex picture and the questions pile up: Why so many different forms with different origins and why then is an Equivalence Principle adopted stating that inertial mass = gravitational? 

• Warum es einfach machen, wenn mann es so schön kompliziert machen kann?

  

Inertial Mass = Gravitational Mass?

This is a continuation of previous posts on operational definition in SI 2019 of mass as gravitational mass. 

In Newton's mechanics inertial mass is exactly equal to gravitational mass as an expression of both (i) universality of free fall and (ii) conservation of energy.  A Universe without (i) and (ii) cannot exist.

In Einstein's mechanics this truly fundamental equality is no longer guaranteed by (i) and (ii), but has to be added as an independent Equivalence Principle EP, which has to be supported by experimental evidence. Accordingly major efforts have been made to find ever more precise experimental confirmation, where the current precision is $10^{-15}$, while new experiments are receiving funding to reach even better precision: 

✅ Best Precision to Date:

• Experiment: MICROSCOPE (CNES, ESA)

• Method: Differential accelerometry in space (free-falling test masses of different materials)

• Result: Difference in acceleration between test masses of different compositions was less than 2 parts in $10^{15}$.

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🚀 Future Target Precisions:

1. Galileo Galilei (GG) satellite – proposed

   • Target: $10^{-17}$

2. STE-QUEST (atom interferometry in space) – proposed

   • Target: $10^{-17}$ (depending on mission configuration)

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To a classical physicist these experiments are similar to precise testing of the validity of the equality 1+1 = 2. 

Einstein presented his Special Theory of Relativity SR in 1905 in a desperate effort to get a university position, and followed up in 1915 with his General Theory of Relativity GR in a desperate attempt to keep the position he managed to get in 1909 under severe criticism of SR. 

Einstein thus took on the role of challenging Newton's mechanics by connecting it to propagation of light which was not mechanics, and by assuming EP as a fundamental postulate of GR as a step away from Newton's mechanics.

Despite severe criticism of both SR and little acceptance of GR before his death in 1955, today Einstein is the undisputed hero of modern physics as the man who showed that Newton was wrong and so opened the door to all sorts of new physics desperately needed after the success of the atom bomb in 1945 had faded.   

The Power of Leibniz' Principle of Sufficient Reason

Leibniz' Principle of Sufficient Reason PSR states:

• Everything must have a reason or a cause.

In other words, there is nothing without reason. It is a powerful principle, but it seems to be forgotten in modern physics, where things can happen without cause as in the basic assumption of quantum mechanics with its wave function demanding a statistical interpretation, because of its multi-dimensionality. The life or death of Schrödinger's cat is supposed to be determined by the throw of a dice thus without reason in direct contradiction to PSR. 

In a previous post PSR was used to show equality of inertial and gravitational mass, which to Einstein was his Equivalence Principle EP assumed as a basic hypothesis of his General Theory of Relativity GR seemingly without reason.

We can use PSR to settle some mysteries of Einstein's Special Theory of Relativity SR concerning the rate of clocks moving with respect to each other with constant velocity (inertial motion). The rate of a mechanical clock like a pendulum cannot depend on inertial motion because the mathematical equations carrying a precise description of the functioning of the clock do not depend on inertial motion (because Newton's 2nd Law does not). There is therefore no reason for clock rates to depend on inertial motion and according to PSR they do not. This means that SR claiming such dependence contradicts PSR. 

As concerns Big Bang, SPR says that since there is no reason for a hot dense initial state to ever exist, it never happened. 

So if we hold on to PSR, then we have to give up SR. We collect:

• PSR implies EP. No need to assume EP as hypothesis as in GR.
• PSR contradicts SR. 
• PSR shows that Big Bang did not happen.

Your conclusion?

Time to Welcome Newton Back

The tragedy of modern physics is replacing Newton's Theory of Gravitation NG from 1687 with Einstein's Theory of Gravitation EG as expressed in Einstein's General Theory of Relativity from 1915. This did not happen over night, but only got momentum after Einstein's death in 1955 into a full victory for Einstein as the established doctrine today embraced by all properly educated physicists.  

Before Einstein NG was viewed to be the jewel of mathematical science all times capturing virtually all of celestial mechanics through one single concise expression in the form of the inverse square law. After Einstein this jewel has lost its shine because EG is viewed to be more precise with NG only a rough first approximation. EG is viewed as a triumph of modern physics widely surpassing NG as outdated classical physics.  

But there are caveats in this success story:

• EG differs from NG to any measurable degree only in very extreme cases, if any.
• NG is computable and serves as a very powerful work horse.
• EG is not computable and so is useless in practice. 

The takeover of EG was prepared by a perceived mystery of NG articulated already from its start by in particular Leibniz as Newton's adversary, namely the apparent presence of instant action at distance in the inverse square law, which could not be explained and so was unthinkable: The gravitational attraction from the Sun is felt without time delay on Earth in its path around the Sun. With even a small delay, the Earth would spin out of orbit together with all the other planets, and this is not what is observed. 

In recent posts I have argued that it is possible to view the inverse square law from a different perspective where instant action-at-distance/fast-global-action is replaced by fast-local-action and slow-global-action, both of which are thinkable. 

Here are details in mathematical form expressing the interplay between gravitational potential $\phi$ as primordial, mass density $\rho$ and $u$ material velocity all depending on a Euclidean space coordinate $x$ and time coordinate $t$ with the dot on top representing differentiation with respect to time:

• $\rho = \Delta\phi$                    (inverse square law in differential form)
• $\Delta\dot\phi =-\nabla\cdot (\rho u)$    (conservation of mass)
• $\rho\dot u=-\vert \rho\vert\nabla\phi$            (Newton's 2nd Law)  

Here $\rho$ is created from $\Delta\phi$ by fast-local-action, changes of $\phi$ in time are created  from changes of $\rho u$ in space by slow-global-action (see previous post) and changes of $u$ from gravitational force $-\nabla\phi$ by fast-local-action. We see 3 concise simple equations in 3 variables $\phi$, $\rho$ and $u$ describing all of celestial mechanics. Amazing! How can you ever imagine to throw away this mathematical model? Compare this post: Unthinkable that the inverse square law is incorrect.

We see the Equivalence Principle EP expressed in Newton's 2nd Law with inertial mass $\rho$ on the left being the same as gravitational mass $\rho$ on the right, which makes Newton's 2nd Law take the form $\dot u=-\nabla\phi$ expressing that all objects independent of mass move the same way subject to gravitation, as experimentally observed and theoretically motivated already by Galileo. We can invoke Leibniz' Principle of Sufficient Reason to understand that inertial mass must be equal to gravitational since a reason for inequality is missing, more precisely a Universe without EP would be only chaos.

Altogether, fast-global-action (unthinkable) is replaced by fast-local-action+ slow-global-action (both thinkable). The main reason to abandon NG for EG can thus be seen to be without solid physical ground, and so Newton can be welcomed back after having been put into the wardrobe since Einstein took over in the 1950-60s.  

If NG in fact does not require fast-global-action, then EG is not needed and modern physics is not what it used to be. What will then a new modern physics with Newton instead of Einstein look like? Definitely simpler than EG and so more understandable than EG and in particular more computable than EG. Nothing but a revolution. A 2nd Scientific Revolution 350 years after the 1st initiated by the Calculus of Newton and Leibniz!

The Equivalence Principle of General Relativity

Bodies of different mass fall the same way.

The pillars of modern physics are quantum mechanics and Einstein's General Theory of Relativity GR, which is based on the following Equivalence Principle EP:

• inertial mass = gravitational mass.                          

In Newtonian Mechanics NM including Newton's Law of Gravitation, EP is not an assumption but instead a necessary consequence of the fact that in NM, as experimentally verified by Galileo dropping different objects from the tower of Pisa while supplying also a theoretical justfication: 

• all bodies independent of mass accelerate the same way in proportion to gravitational force.  

More precisely, with A acceleration and F gravitational force per unit mass, in NM we have 

• A = F

which we can write 

• MA = MF

where M is mass, as an expression of Newton's 2nd Law: mass x acceleration = force. We see that $M$ appears on both sides as an expression of EP. 

Summary: EP is automatically satisfied in NM. It does not make sense in NM to introduce EP as an assumption. On the other hand, EP is an assumption of GR. EP is self-evident in NM but not so in GR, where (P) is made into assumption as if it is possible that inertial mass is different from gravitational mass. This makes GR into a strange perturbation of NM with the following question: What makes GR different from NM when they share the same EP? 

ChatGPT-4: Truth about the Equivalence Principle: Self-Evident!

This expresses that if A is true then B is true and vice versa, that is that A and B are equivalent.

This is a continuation of the previous post revealing with the help of ChatGPT-4 that the Principle of Relativity is simply an agreement empty of real physics. We now ask GPT-4 if the same is true about the Principle of Equivalence and get the following response: 

Me: Is it true that gravitational mass reflects reaction/acceleration to gravitational force?

GPT-4: Yes, that is correct.

Me: Is it true that inertial mass reflects reaction/acceleration to force of any kind?

GPT-4: Yes, that is correct.

Me: Since gravitational force is a force and inertial mass reflects reaction/acceleration to any force, does it mean that inertial mass is reaction/acceleration to gravitational force?

GPT-4: Yes, that is correct.

Me: Doesn’t that mean that gravitational mass is equal to inertial mass?

GPT-4: Yes, exactly. According to the equivalence principle in physics, gravitational mass and inertial mass are considered to be equal. This principle states that the gravitational mass of an object, which determines its response to gravitational forces, is equivalent to its inertial mass, which determines its resistance to changes in motion.

Ok, so again GPT-4 helps us to uncover that the supposed deep mysteries of real physics expressed by the Principles of Relativity and Equivalence as the basic assumptions of Einstein's Special/General Theory of Relativity, in fact only represent certain agreements and so are empty of real physics. This is something a living physicist would never admit, only a naive GPT-4 not yet educated enough.  

Reflection: The concept of equivalence (denoted $\Leftrightarrow$) is different from identity (denoted $\equiv$). $A\Leftrightarrow B$ thus expresses that certain aspects of $A$ and $B$ are the same, but they are not necessarily the same in every respect, thus not necessarily identically the same. An example is the Fundamental Theorem of Calculus which reads: If $F(x)$ is a primitive function of $f(x)$ so that $\frac{dF}{dx}=f$, then 

• $\int_a^bf(x)\, dx = F(b) - F(a)$,        (*)

which expresses that the $\int_a^bf(x)\, dx$ and $F(a) - F(b)$ take the same numerical value, but are computed in different ways with the integral corresponding to a summation of increments $f(x)\, dx$ and

$F(b)-F(a)$ a difference of primitive function values, and thus in a way can be viewed to not be identically the same. The other way around, (*) can be made into an identity if it is used to define the primitive function $F(x)$ in terms of the integrand $f(x)$, whereby the Fundamental Theorem instead expresses that $\frac{dF}{dx}=f$ as an equivalence. In any case, (*) expresses equivalence but not identity. 

 

We compare with $E=mc^2$ supposed to express mass-energy equivalence, or in Einstein's words different manifestations of the same thing. With our above experience from the Principles of Relativity and Equivalence we are led to suspect that this is also an empty agreement simply defining energy $E$ in terms of mass $m$. 

Equivalence Principle: Einstein vs Galileo/Newton

General Theory of Relativity is based on Einstein's Equivalence Principle EEP, to be compared with the Galileo/Newton's Equivalence Principle expressing Universality of Free Fall UFF.

UFF means that matter independent of composition reacts the same way in a gravitational field and so shows the same free fall. UFF is a prediction of Newtonian mechanics verified with very high precision starting with Galileo's free fall experiments from the Tower of Pisa. The proof that UFF is a consequence/prediction of Newton's mechanics is given in two lines here. Check it out.

Einstein formulated EEP considering two reference frames, K placed in a uniform gravitational field and K′ uniformly accelerated, as follows:

• As long as we restrict ourselves to pure mechanical processes in the realm where Newton’s mechanics holds sway, we are certain of the equivalence of the systems K and K′. 
• But this view of ours will not have any deeper significance unless the systems K and K′ are equivalent with respect to all physical processes, that is, unless the laws of nature with respect to K are in entire agreement with those with respect to K′. 
• By assuming this to be so, we arrive at a principle which, if it is really true, has great heuristic importance. 
• By theoretical consideration of processes which take place relatively to a system of reference with uniform acceleration, we obtain information as to the career of processes in a homogeneous gravitational field.

Einstein thus acknowledges UFF to be a consequence of Newton's mechanics, but finds a need to extend it beyond Newton's mechanics to all physical processes, and then things get complicated. So complicated that Einstein must leave the safe haven of Newton's mechanics with UFF secured, and resort to assuming EEP to be a postulate, now asking for experimental conformation. But that is virtually impossible, since all physical processes are involved. 

So there we stand today: UFF is secured by Galileo/Newton. EEP is waiting for confirmation. 

PS1 UFF is also referred to as the weak equivalence principle, and EEP as the strong equivalence principle.

PS2 Note that UFF does not exclude internal motion of a body subject to a non-uniform gravitational field (tidal motion). 

PS3 Einstein's bright idea was to ask what may happen to Newton's mechanics if it is extended beyond its realm of mechanics to electromagnetics with material bodies traveling at near the speed of light, however  without connection to reality. There is no reason to question Newton's mechanics, and anyway doing that is stupid.   

Why Inertial Mass = Gravitational Mass

In equilibrium the spring force = gravitational force/weight.

The Equivalence Principle EP states that: 

• Inertial Mass = Gravitational Mass.  

EP means that all material bodies react the same way to gravitational force as to other forces, or specifically that they react to inertial force the same way as to gravitational force. In Newton's mechanics this follows from the fact that the size of any force ultimately is determined by comparison with gravitational force. 

For example, the force exerted by an elastic spring under extension is determined by hanging different bodies in the spring and measuring the extension, see above picture. This means that in Newtonian mechanics the gravitational force is used as reference for all types of force and so EP is valid by definition/agreement. Gravitation is the common denominator. 

In Newtonian mechanics it is thus meaningless to seek to verify EP by an experiment, and senseless to seek to do that in a very expensive experiment, like the MICROSCOPE.  It would be the same as to seek to experimentally verify that there are 100 centimeters on a 1 meter. Most people would say that this would not be meaningful, even if very high precision could be reached.  

EP is thus a consequence of Newton's mechanics, but it is different in Einstein's mechanics, where EP in instead serves as a postulate, which then in principle can be subject to experimental testing. This is what was reported in the previous post: A very high-precision experiment confirming EP! Like testing that there are 100 centimeters on 1 meter, to very high precsision? 

On top of that the experiment is supposed to verify not only EP as postulate of Einstein's mechanics (General Theory of Relativity GR), but also serve as a confirmation of the profound revolution brought to humanity by GR. This is mind-boggling. If EP is nothing but an agreement, then GR would also simply be an agreement, and so not real science.

The very high-precision of the experimental verification of EP supports a suspicion that what is tested is in fact a definition/agreement rather than a law/theory of physics. The consequences as concerns the physicality of GR are possibly far-reaching...

Compare with the low precision of the value of the gravitational constant $G$: less than 5 decimals. 

Recall that in the new SI units standard, agreements take the role of laws of physics when defining meter in terms of prescribed speed of light, and kg in terms of prescribed equivalence with energy. However, this is not anything which modern physicists want to agree on: the prescriptions/agreements are also laws of physics! The Best of Worlds! Compare with posts on SI 2019 standard.

Corruption of Modern Physics 8: Equivalence Principle

In Dialogue Concerning the Two Chief World Systems Galileo lets Salviato explain to Simplicius why different bodies with different masses show the same free fall (in vacuum without air resistance) thus with the same acceleration subject to a gravitational force, in other words that inertial mass scaling with  acceleration is the same as gravitational mass scaling with gravitational force. 

Galileo demonstrated this fact by dropping different objects from the Tower of Pisa. Galileo argued that this must be so because if different parts of a body in free fall would fall differently, bodies would be ripped apart by internal forces and the World could not exist. This is called Galileo's Equivalence Principle and is a cornerstone of Newtonian Mechanics. It says that there is only one type of mass, namely inertial mass, which manifests itself as force resistance to acceleration expressed by Newton's 2nd Law.  If you want to find the mass of a body, measure the force required to put it into motion.  

It is now reported from Institute of Theoretical Physics, University of Paris-Saclay, France, that 
the 

• MICROSCOPE Satellite Experiment has tested the equivalence principle with an unprecedented level of precision.

This is simply Galileo's experiment again now with higher precision. No surprise. So what is the meaning of this apparently ground breaking experiment? Well, it is presented as evidence that Einstein's General Theory of Relativity with a new explanation of gravitation, is correct. 

How can that be when the Equivalence Principle is a cornerstone of Newton's theory of gravitation? How can an experiment confirming Newton's theory be used to show that also General Relativity supposed to be different from Newton's theory, is correct? How can the same evidence be used to support both a basic theory like Newton's and a generalisation thereof like Einstein's supposed to be different? 

Experimentally evidence that Einstein's theory is better than Newton's, would require an experiment which agrees with Einstein's theory but not Newton's. An experiment confirming Newton's theory cannot serve this purpose. Yet the objective of the MICROSCOPE Satellite Experiment is just this! As an example of what modern physicists in the spirit of Einstein do today to advance physics.  

Gravitational Mass = Inertial Mass: Einstein or Galileo?

Einstein takes as postulate of his general theory of relativity that

• gravitational mass = inertial mass.  

To Einstein this represented a deep insight into the inner nature of things, which he named the Equivalence Principle. To Galileo the same thing was a most natural consequence of his theoretical insight from experiments of dropping objects from the Tower of Pisa noting that all objects fall in the same way (modulo air resistance) and reflecting over the connection between force and motion.

Let us see if we can understand what to Einstein was beyond comprehension and to Galileo more or less self-evident. Newton's second law states that 

• $m_i\frac{dv_i}{dt}=F_i$, 

where $m_i$ is the inertial mass of a body showing acceleration $\frac{dv_i}{dt}$ with $v_i$ velocity and $t$ time when subject to a force $F_i$. On the other hand, the same body when subject to a gravitational force $F_g$, shows an acceleration $\frac{dv_g}{dt}$ satisfying 

•   $m_g\frac{dv_g}{dt}=F_g$,

where $m_g$ is the gravitational mass. 

To find out if $m_i=m_g$, let us consider the following experiment: Consider two identical bodies,  a body $A$ at rest on a frictionless table and another body $B$ in your hand with the two bodies connected by a weightless string stretched over a frictionless wheel attached at the end of the table, see picture in earlier version of this post. Then remove your hand and observe the action of the two-body-string system.  Observe that $A$ is acted upon by the horisontal string force $F_s$, while $B$ is acted upon by $F_g-F_s$ with $F_g$ the gravitational force acting on $B$. Since A and B have the same acceleration, we have

• $\frac{F_s}{m_i}=\frac{F_g-F_s}{m_g}$.  

If we now observe that 

• $F_s=\frac{F_g}{2}$,    (1)

then we can conclude that $m_g=m_i$ as a simple experimental verification of the Equivalence Principle. We can also argue that (1) must be true according to Leibniz' principle of sufficient reason, since there is no reason that the two-body-string system should not show this form of symmetry. We can also argue that (1) must hold if we re-orient the system to be all horisontal and pull $B$ with a certain force $F$ which must result in a string force $\frac{F}{2}$. 

Summing up, we have given simple evidence that gravitational mass = inertial mass, based on the insight that there is only one type of mass, namely inertial mass as a measure of acceleration vs force. Since gravitation is a force the measure of acceleration vs gravitational force as gravitational mass is necessarily the same as inertial mass. This is captured in the experiment with $B$ subject to (vertical) gravitational force (minus vertical string force) and $A$ to horizontal string force. 

The Equivalence Principle is thus a direct consequence of Newtonian mechanics, and as such a most questionable empty Basic Postulate for general relativity. As usual Einstein managed to create confusion rather than clarification. For more reason to this verdict see earlier posts on the Equivalence Principle.    

Equivalence Principle as Definition: Experiment

Here is little experiment you can do yourself on the kitchen table supporting the idea that intertial mass is made equal to gravitational mass by definition as a result of a definition of force in terms of gravitational force:

Take two identical pieces of material, put one of the pieces on a horisontal table with frictionless surface and connect the other by a weightless rope as indicated in the picture and let go from rest.

Record the acceleration of the system. Observe that it is half of the gravitational acceleration of one of the pieces in free fall. Conclude that inertial mass = gravitational mass and that force ultimately is defined in terms of gravitational force as expressed by the green arrows.

Understand that what you test with the experiment is if mass = force/acceleration is independent of orientation of Euclidean coordinate system.

Gravitational Mass = Inertial Mass by Definition: Hard Thinking

                                 Typical illustration of equivalence principle. Get the point?

In Newtonian mechanics as already observed and understood by Galieo, inertial and gravitational (heavy) mass are equal, because there is only one form of mass and that is inertial mass as a measure of acceleration vs force per unit of volume. Since Newtonian gravitation is a force per unit of volume, gravitational mass is equal to inertial mass, by definition, as expressed by the fact that the dimension of gravitation is $m/s^2$. See also Chap 18 of Many-Minds Relativity.

Let us compare this insight with what modern physics says as told by Nigel Calder in Magic Universe:

• A succession of experiments to check the equivalence principle—the crucial proposition that everything falls at the same rate—began with Lorand Eötvös in Budapest in 1889. After a century of further effort, physicists had improved on his accuracy by a factor of 10,000. The advent of spaceflight held out the possibility of a further improvement by a factor of a million.
• If another theory of gravity is to replace Einstein’s, the equivalence principle cannot be exactly correct. Even though it’s casually implicit for every high-school student in Newton’s mathematics, Einstein himself thought the equivalence principle deeply mysterious. ‘Mass,’ he wrote, ‘is defined by the resistance that a body opposes to its acceleration (inert mass). It is also measured by the weight of the body (heavy mass). That these two radically different definitions lead to the same value for the mass of a body is, in itself, an astonishing fact.’
• Francis Everitt of Stanford put it more forcibly. ‘In truth, the equivalence principle is the weirdest apparent fact in all of physics,’ he said. ‘Have you noticed that when a physicist calls something a principle, he means something he believes with total conviction but doesn’t in the slightest degree understand.’
• Together with Paul Worden of Stanford and Tim Sumner of Imperial College London, Everitt spent decades prodding space agencies to do something about it. Eventually they got the go-ahead for a satellite called STEP to go into orbit around the Earth in 2007. As a joint US–European project, the Satellite Test of the Equivalence Principle (to unpack the acronym) creates, in effect, a tower of Pisa as big as the Earth. Supersensitive equipment will look for very slight differences in the behaviour of eight test masses made of different materials— niobium, platinum-iridium and beryllium—as they repeatedly fall from one side of the Earth to the other, aboard the satellite.
• ‘The intriguing thing,’ Everitt said, ‘is that this advance brings us into new theoretical territory where there are solid reasons for expecting a breakdown of equivalence. A violation would mean the discovery of a new force of Nature. Alternatively, if equivalence still holds at a part in a billion billion, the theorists who are trying to get beyond Einstein will have some more hard thinking to do.’ 

So Einstein thought to be deeply mysterious, what every high school student directly understands, and was able to imprint his idea into the brains of all modern physicists, who now have some hard thinking to do...

Einstein skillfully jumped between definition as a tautology true by construction and physical principle/law, which may be valid/true or not, thereby creating a total confusion. Another aspect is the constancy of the speed of light, which today is used as definition with the meter defined by distance traveled by light in certain time, yet physicists go around and believe that this works because the speed of light is constant. If you cannot distinguish between a definition without content and statement with content, then you may find yourself in trouble and mislead others...

PS This previous post may be consulted: The Principal Difference between Principles and Laws in Physics. Note in particular the distinction that a law is typically expressed as a formula, while a principle is expressed in words e.g. as equality of inertial and gravitational mass.

Einsteins "Scientific Method": Magic Physics from Definition

Einstein "scientific method", which brought him immense fame as the greatest physicist of all times, consists of:

• Start from a definition, convention or stipulation/law without physical content, and then draw far-reaching consequences about the physics of the world.

It is not hard to understand that such a "method" cannot work: You cannot draw meaningful conclusions about the world simply from a definition empty of physics content. You cannot develop a meaningful scientific theory from a definition that there are 100 centimeters on a meter. 

Einstein cleverly covered up by naming his definitions or conventions or stipulations/laws, "principles":

1. Equivalence Principle: Gravitational mass is equal to inertial mass.
2. Relativity Principle: Observations in inertial coordinate systems moving with constant velocity with respect to each other, are to be connected by the Lorentz transformation.
3. Covariance Principle: Physical laws are to have the same form independent of the choice of coordinate system.

Here 1. is an empty definition, because there is only one mass, and that is inertial mass, which measures acceleration vs force and gravitational force is a force. Gravitational mass is equal to inertial mass by definition. Attempts to "prove/verify" this experimentally, which are constantly being made with ever increasing precision and always with the same result of equality, are as meaningful as experiments attempting to verify that there are 100 centimeters on a meter, which could very well be the next grand challenge for LHC, in the spirit of Einstein.

2.  stipulates that different physical phenomena are to be viewed to be the same. This is because the Lorentz transformation is not invariant with respect to initial conditions, and thus Einstein stipulates that  two waves satisfying the same form of wave equation, but having different initial conditions, shall be viewed to be the same. No wonder that with this play with identities, all sort of strange effects of time dilation and space contraction can be drawn out of  a magicians hat.

It is clear that physical laws in general take different forms in different coordinate systems, and thus 3. is an absurd stipulation. Alternatively, it is trivial and just says that a physical law will have to transform when expressed in different coordinates so that the law has the same physical content. So 3. is either absurd or trivial, in both cases devoid of physics.

It is depressing that none of this can be understood by leading modern physicists. Nada. Even more depressing is that the discussion is closed since 100 years.

The Equivalence Principle from Newton's 2nd Law

We now continue New View of Motion under Gravitation without Classical Mysteries with a study of the Equivalence Principle (EP) stating that inertial mass is equal to gravitational  (heavy) mass, or in other words, that all matter independent of mass and composition reacts the same way to gravitation. EP was identified by Galileo and by Newton made into a corner stone of his theory of gravitation.

The question is if EP is a deep mystery or a necessity which can be understood? Why does a stone and a feather fall the same way when dropped from the Tower of Pisa (assuming no air resistance)? Is it mystery or not?

We recall our Newtonian model of matter subject to gravitation expressed by the equations (to be satisfied for all $x$ and positive time $t$):

• $\rho (x,t) =\Delta\phi (x,t)$                              (Newton's law of gravitation)
• $\ddot x(t)= -\nabla\phi (x(t),t)$                          (Newton's 2nd law in math form)
• $\Delta\dot\phi +\nabla\cdot (u\Delta\phi )=0$                        (Evolution equation for $\phi$),

where $\phi$ is a gravitational potential $\phi (x,t)$ depending on a Euclidean space coordinate $x$ and time $t$, and $\rho (x,t)$ is matter density and $x(t)$ denotes a trajectory with acceleration $\ddot x(t)$ and velocity $\dot x(t)=u(x(t),t)$ specified by initial conditions $x(0)$ and $\ddot x(0)$, with the dot denoting differentiation with respect to time. 

We thus consider the gravitational potential $\phi$ as the primordial object  and the matter density $\rho (x,t)$ as a derived physical quantity given by Newton's law of gravitation: $\rho (x,t) =\Delta\phi (x,t)$. 

We consider trajectories $x(t)$ as mathematical constructs given by Newton's 2nd law in its mathematical form: $\ddot x(t)= -\nabla\phi (x(t),t)$. Newton's 2nd law in math from states a connection between motion $\ddot x(t)$ and the gradient of the gravitational potential $\nabla\phi (x,t)$, which can be viewed to reflect a balance between two types of energy: 

• KE(t) = $\frac{\vert\dot x(t)\vert^2}{2}$                                              (kinetic energy) 
• GE(t) = $\phi (x(t),t)$,                                     (gravitational energy):

with

• $\frac{d}{dt}(KE + GE)=0$. 

We view $\rho (x,t) =\Delta\phi (x,t)$ as expressing local "creation of matter" with density $\rho (x,t)$ by "action" of the Laplacian differential operator $\Delta$ operating on the gravitational potential $\phi (x,t)$ locally in space and time. We note that the "creation of matter" does not depend on the nature or composition of matter, and we can thus view the created matter to be a form of primordial matter with only quality being density $\rho$ from which all matter with different nature and composition is formed. 

Suppose now we multiply Newton's 2nd law by $\rho (x,t)$ to get

• $\rho\ddot x(t)= -\rho (x,t)\nabla\phi (x(t),t)$,              (Newton's law in physics form)    

and make the following physical interpretation

• $F =-\rho (x,t)\nabla\phi (x(t),t)$ is gravitational force
• $\ddot x(t)$ is material acceleration
• $\rho (x,t)$ is matter density.

We can then view Newton's 2nd law in physics form to express EP in the form:

• Material motion under gravitation is independent of matter nature, composition and density.

We sum up:

• EP is a consequence of Newton's 2nd law.
• Newton's 2nd law serves to maintain a balance between kinetic energy depending on motion and gravitational energy depending on position.  

The essence given by the Creator is thus a Universe with a dynamics in time based on a balance between two forms of energy, kinetic energy of motion and gravitational energy of position, maintained by Newton's 2nd law. In this Universe, EP is valid as a consequence of Newton's 2nd law. 

Defining non-gravitational forces $F(x(t),t)$ by Newton's law in the form 

• $F(x(t),t) \equiv \rho(x(t),t)\ddot x(t) + \rho (x(t),t)\nabla\phi (x(t),t)$,

extends this Universe to effects of forces other than gravitation. 

Einstein's general theory of relativity is supposedly based on EP. If now EP is a consequence of Newton's 2nd law, we are led to the conclusion that Einstein's mechanics is no different from Newton's. 

Comparison with Standard Newtonian Mechanics

If we now accept that EP is a consequence of Newton's 2nd law, it is natural to ask in what sense the New View may be more illuminating than the standard view?

In standard Newtonian mechanics, matter with density $\rho (x,t)$ is primordial with the gravitational  potential $\phi$ being generated from $\rho$ as a solution of $\Delta\phi =\rho$ (requiring instant action at distance). The gravitational force acting on a lump of matter of density $\rho$ will then be given by $\rho\nabla\phi$ and EP will then, as above, be equivalent to Newton's law in the form $\rho\ddot x + \rho\nabla\phi=0$. 

In this case matter of density $\rho$ appears in three forms: (i) as generator of gravitation in $\Delta\phi =\rho$, (ii) as gravitational mass in $\rho\nabla\phi$ and (iii) as inertial mass in $\rho\ddot x$,  all supposed to be the same, which may be viewed as coincidence or as a mystery. 

In the New View, $\rho =\Delta\phi$ appears in only one form as a product of $\phi$ combined with Newton's 2nd law in math form expressing a balance of kinetic and potential energies, and the equivalence of this form of mass with gravitational and inertial mass is simply a matter of definition and thus no mystery. 

The New View thus opens to understanding that EP is rather a matter of definition, which is true by the construction of language,  than a statement about physics, which may be true or false. In particular, it points to obvious difficulty of avoiding circular reasoning when seeking to test EP experimentally.