Are Physical Quantities Additive?

[Paul Talbot]

The Non-Additivity of Speeds

The non-additivity of speeds (or velocities) is a physical property that has been demonstrated and verified experimentally. Indeed, if speeds were additive, then the sum of certain values could exceed the maximum of this quantity, that is the speed of light (c).
For easier reading, I will call the relativistic composition of velocities «adjunction» and I will note this operation ⨤ : «almost plus, or adjunct to».
If a and b are two speeds expressed in maximum units (c = 1), then the equivalence between adjunction and addition is:
a ⨤ b <=> (a+b) / (1+ab)
For example:
0.5 ⨤ 0.5 <=> (0.5 + 0.5) / (1 + 0.5 · 0.5) = 1/1.25 = 0.8
and not 1, as suggested by the addition. This equivalence results from the application of the Lorentz transformations, which are used by special relativity. If different units are used, the equivalence between adjunction and addition remains, but involves the speed of light (c):
a ⨤ b <=> (a+b) / (1+ab/c^2)
When the velocities involved are much smaller than that of light, the result of the adjunction (⨤) is very close to that of the addition (+). In fact, the result of the adjunction is always less than or equal to that of the addition, as well as to the value of the maximum speed (c).
Except for Speed, it is generally assumed that other physical quantities, such as Length and Time, are additive. This is indeed what is observed at the human scale, but how to know if this assumption remains valid at the cosmological scale?

The Non-Additivity of Length and Time

Surprisingly, it can be shown that Length and Time are not additive.
If lengths were additive, then physically, we should observe:
• (1 m + 1 m) = 2 m
By traveling these distances in one second, we should observe:
• (1 m + 1 m) / 1 s = 2 m / 1 s
The division being distributive with respect to the addition, we should observe:
• (1 m / 1 s) + (1 m / 1 s) = 2 m / 1 s, which can be written:
• 1 m/s + 1 m/s = 2 m/s
Now, this is not what we observe. The latter equation is physically inaccurate because it does not use the correct law of composition (⨤). The difference is tiny, but according to the above, we calculate that 1 m/s ⨤ 1 m/s = (1+1) / (1+1/c2) m/s ≈ 1.999 999 999 999 999 978 m/s.

Similarly, if we assume that Time is additive, then physically, we should observe:
• (1 s + 1 s) = 2 s
Accelerating by 1 m/s^2 during these durations, we should observe:
• 1 m/s^2 · (1 s + 1 s) = 1 m/s^2 · 2 s
The multiplication being distributive with respect to the addition, we should observe:
• (1 m/s^2 · 1 s) + (1 m/s^2 · 1 s) = 1 m/s^2 · 2 s
This corresponds to the addition of velocities which is physically inaccurate as previously described:
• 1 m/s + 1 m/s = 2 m/s
These equations show that the additivity of Length or Time implies that of Speed. Since we observe that Speed is not additive, Length and Time would not be additive either. This is called a demonstration by reduction to the absurd.

One might think that the distribution of division or multiplication should not be allowed, because it transforms the addition of lengths or durations into the addition of speeds. This prohibition would automatically imply that Length and Time are not additive. In mathematics, the product of (a) by the sum (b + c) is always equal to the sum of the products (ab + ac) :
a · (b + c) = ab + ac
This is a fundamental property of addition. If this property is not verified, it means that the composition of the involved quantities is not a true addition, even if the result is close to it. This is exactly what is observed in the case of Length and Time.
If you believe that these demonstrations are wrong, would you please tell me where the reasoning is wrong?
Thank you for reading.

[nemos]

I could be lying, but if I remember correctly:
Relativistic relationships determine the dependence of length/mass/time on velocity. If velocity is variable then these quantities are also variable and consequently their sums are determined as functions of velocity, but if velocity is constant, or is considered at a certain instant at which it can be taken as constant the mathematical operations are completely Newtonian.

[alan1000]

This is scientific philosophy as it should be! My own instinctive response to the initial post was that it did not differentiate the Newtonian and the Relativistic in a sufficiently rigorous way, but the first response expresses the point more succinctly than ever I could have.