Non-Newtonian Mathematics Instead of Non-Newtonian Physics: Dark Matter and Dark Energy from a Mismatch of Arithmetics
Authors: Czachor Marek 
Published: July 2020

Newtonian physics is based on Newtonian calculus applied to Newtonian dynamics. New paradigms such as ‘modified Newtonian dynamics’ (MOND) change the dynamics, but do not alter the calculus.
However, calculus is dependent on arithmetic, that is the ways we add and multiply numbers. For example, in special relativity we add and subtract velocities by means of addition $$\beta _1\oplus \beta _2=\tanh \big (\tanh ^{-1}(\beta _1)+\tanh ^{-1}(\beta _2)\big )$$

β
1

⊕

β
2

=
tanh

(

tanh

-
1

(

β
1

)

+

tanh

-
1

(

β
2

)

)

, although multiplication $$\beta _1\odot \beta _2=\tanh \big (\tanh ^{-1}(\beta _1)\cdot \tanh ^{-1}(\beta _2)\big )$$

β
1

⊙

β
2

=
tanh

(

tanh

-
1

(

β
1

)

·

tanh

-
1

(

β
2

)

)

, and division $$\beta _1\oslash \beta _2=\tanh \big (\tanh ^{-1}(\beta _1)/\tanh ^{-1}(\beta _2)\big )$$

β
1

⊘

β
2

=
tanh

(

tanh

-
1

(

β
1

)

/

tanh

-
1

(

β
2

)

)

do not seem to appear in the literature. The map $$f_{\mathbb{X}}(\beta )=\tanh ^{-1}(\beta )$$

f
X

(
β
)

=

tanh

-
1

(
β
)

defines an isomorphism of the arithmetic in $${\mathbb{X}}=(-1,1)$$

X
=
(
-
1
,
1
)

with the standard one in $${\mathbb{R}}$$
R
. The new arithmetic is projective and non-Diophantine in the sense of Burgin (Uspekhi Matematicheskich Nauk 32:209–210 (in Russian), 1977), while ultrarelativistic velocities are super-large in the sense of Kolmogorov (Technika Molodezhi 10:16–19 (11:30–33 in Russian), 1961). Velocity of light plays a role of non-Diophantine infinity. The new arithmetic allows us to define the corresponding derivative and integral, and thus a new calculus which is non-Newtonian in the sense of Grossman and Katz (Non-Newtonian calculus, Lee Press, Pigeon Cove, 1972). Treating the above example as a paradigm, we ask what can be said about the set $${\mathbb{X}}$$
X
of ‘real numbers’, and the isomorphism $$f_{{\mathbb{X}}}:{\mathbb{X}}\rightarrow {\mathbb{R}}$$

f
X

:
X
→
R

, if we assume the standard form of Newtonian mechanics and general relativity (formulated by means of the new calculus) but demand agreement with astrophysical observations. It turns out that the observable accelerated expansion of the Universe can be reconstructed with zero cosmological constant if $$f_{\mathbb{X}}(t/t_H)\approx 0.8\sinh (t-t_1)/(0.8\, t_H)$$

f
X

(
t
/

t
H

)

≈
0.8
sinh

(
t
-

t
1

)

/

(
0.8

t
H

)

. The resulting non-Newtonian model is exactly equivalent to the standard Newtonian one with $$\Omega _\Lambda =0.7$$

Ω
Λ

=
0.7

, $$\Omega _M=0.3$$

Ω
M

=
0.3

. Asymptotically flat rotation curves are obtained if ‘zero’, the neutral element $$0_{{\mathbb{X}}}$$

0
X

of addition, is nonzero from the point of view of the standard arithmetic of $${\mathbb{R}}$$
R
. This implies $$f^{-1}_{{\mathbb{X}}}(0)=0_{{\mathbb{X}}}>0$$

f
X

-
1

(
0
)

=

0
X

>
0

. The opposition Diophantine versus non-Diophantine, or Newtonian versus non-Newtonian, is an arithmetic analogue of Euclidean versus non-Euclidean in geometry. We do not yet know if the proposed generalization ultimately removes any need of dark matter, but it will certainly change estimates of its parameters. Physics of the dark universe seems to be both geometry and arithmetic.