Mach's principle proposes that inertia is an interaction between a body and the rest of the mass in the universe. The document presents Mach's principle, discusses how it could explain the origin of inertia by deriving kinetic energy as an interaction energy between bodies. It then shows calculations modeling the universe as a spherical shell to derive an expression for a particle's total interaction energy with the universe that is equivalent to its inertial mass.

1. A Quick Introduction to
Mach’s Principle
A pre-relativistic account to the origin of
inertia…
ADEMIR XAVIER JR
NOVEMBER 2017.
BRAZILIAN SPACE AGENCY (XAVNET2@GMAIL.COM)

2. A State of conflict
Space and time were originally formulated by Newton in terms
of absolute concepts.
Leibnitz, Berkeley: one can only speak about space in relation
to things, and not in relation to the “emptiness” of space.
Ernst Mach (1883): “The Science of Mechanics”: has deeply
influenced Einstein in the development of relativity.
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ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.

3. Illustration 1
3
“Space”
The state of movement of a body “particle” in the infinite void. Which movement ?
ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.

4. Illustration 2
4
Space
State of movement of a body “particle” in space in relation to another body.
There is motion in relation to that other particular body...
ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.

5. Mach’s principle
Since the state of motion can only be described with reference to other
bodies, the interaction energy should be in the form:
5
r
The energy of interaction between two bodies can only depend on the relative
position of these bodies or/and the superior time derivatives of their separating
distance...
ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.

6. Energy of interaction
A given body energy is the total energy associated to it and
nothing more. We know from mechanics that this energy is
given by
𝐸 =
1
2
𝑚𝑣2 + 𝑈(𝑟)
Here mv2/2 is the kinetic term. The aim of Mach’s principle is to
retrieve this term as an “interaction energy” of the body with
“the rest of the Universe”.
Thus, inertia is a consequence of an underlying more
fundamental interaction and not a primary cause.
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ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.

7. v
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ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.
Distant galaxies
Interaction energy

8. “Implementing” Mach’s principle
As a “principle” it cannot be proven, but only admitted.
Inertia is the result of the potential “generalized” energy of
interaction among a body and the “fixed” stars (represented
today as the mass of distance galaxies).
Action-at-a-distance is needed to obtain inertia and the
following assumptions:
◦ The total sum of forces upon a body is zero. Terms of inertia are
obtained by assuming a certain interaction force between the body and
each particle in the Universe.
◦ The total “generalized” energy of a body is constant (energy
conservation).
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ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.

9. Inertial and gravitational mass
As a “coincidence”, the factor “m” which appears in the term
mv2/2 is numerically the same (by several decimal places) as
the “m” entering the gravitational potential energy:
𝐺
𝑚
𝑟
Mach’s principle would explain the coincidence in the
numerical values of both masses in a natural way: they have
the same origin as an underlying relational energy.
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ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.

10. An “ansatz” interaction
potential
A suggestive interaction energy between two particles of mass  and ’ is:
Justification: the kinetic energy results from the interaction energy
and depends only on the masses and distances between the two
particles. Energy should be proportional to the square of the relative
velocity between the particles. The energy should fall with distance...
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ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.
r

’
 E. Schrödinger. Die Erfüllbarkeit der Relativtätsforderung in der Klassischen Mechanik,
Annalen der Physik, 77, p. 325 (1925)

11. Calculating the interaction energy of a particle with
the universe
Universe represented as a
spherical shell where a particle
is. One should sum the
contributions of several
infinitesimal elements of the
shell as shown.
The shell radius is R, and the
shell mass density is .
Assuming 𝜌 ≪ 𝑅, we can take
𝜃 = 0 exploring the symmetry.
Hence 𝑟 ≈ 𝑅.
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
d’
(ρ, = 0,)
(R,’,’)
R
Uniform mass
distribution density 
ρ
ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.
Particle of mass  distant 
from the shell “center”.
r

12. Calculating the interaction energy of a particle
with the universe
Hence:
The total interaction energy will be
or
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ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.

13. Analyzing results
One can call
But
So
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Radial “kinetic” term Energy term
corresponding giving
rise to the “centrifugal
force”

ρ
Inertial mass
Newton’s gravitational
constant
ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.

14. In summary
The total interaction energy of a particle with the Universe is:
Which is obtained from a more “fundamental” interaction energy of the particle with all
parcels of the Universe represented as a spherical shell. This fundamental interaction is given
by
In such a way that the gravity strength and its constant of proportionality is linked to the
Universe through
That is, the Universe’s mass and radius.
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ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.

15. m’
m
Other consequences
The precession of the perihelion of Mercury: is a consequence of
an “excess” of radial inertia...
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The rate of precession as a function of the sun’s mass (m’)
and orbit geometry (semi major and minor axis, period of
revolution etc)
ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.

16. Issues...
The interaction is via action-at-a-distance or “instantaneous
action”.
But Einstein strongly believed in the interactive origin of inertia.
He also believed it would be possible to implement Mach’s
principle in relativity. Interestingly enough, for E. Schrödinger
(1925) this would be possible too:
“But, in the same way we can interpret the seemingly minimal
and instantaneous influence of a distance star upon a pendulum
on Earth through the introduction of gravity propagation at light
velocity, it seems to me possible to make the calculations of the
terms depending on dr/dt without violating the propagation of
light at limited speed. This will be true provided we can find a
relation that renders irrelevant in the calculations the state of
motion, be it instantaneous our delayed in accordance to a given
propagation dependent time interval.”
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ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.

17. Some past references on this matter
E. Schrödinger. Die Erfüllbarkeit der Relativtätsforderung in der Klassischen
Mechanik, Annalen der Physik, 77, p. 325 (1925)
A. K. T. Assis, On Mach's principle, Foundations of Physics Letters, Vol. 2, pp. 301-
318 (1989);
A. L. Xavier Jr. and A. K. T. Assis, O cumprimento do postulado de relatividade na
mecânica clássica - Uma tradução comentada de um texto de Erwin Schrödinger
sobre o princípio de Mach, Revista da Sociedade Brasileira de História da Ciência,
Vol. 12, pp. 3-18 (1994);
 A. L. Xavier Jr. and A. K. T. Assis, Schrödinger, Reissner, Weber e o princípio de
Mach, Revista da Sociedade Brasileira de História da Ciência, Vol. 17, pp. 103-106
(1997);
A. K. T. Assis e O. Pessoa Jr., Erwin Schrödinger e o princípio de Mach, Cadernos de
História e Filosofia da Ciência, Vol. 11, pp. 131-152 (2001).
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ADEMIR L. XAVIER JR (NOVEMBER 2017). QUICK INTRODUCTION TO
MACH'S PRINCIPLE. AEB.