What is Equality in Physics?

In modern western society the concepts of equity or equality plays an important role. Equality can refer to equality of possibilities or equality of outcomes in a democratic system or equality of everybody in an Orwellian sense in a communistic society. Equity can refer to equality over different gender/sex and can connect to an idea of equality woman = man and man = woman.

We also find extreme forms of equality in 1984: War is Peace, Slavery is Freedom....

We understand that equality can express both a truth and serve as a disguise or cover-up of a truth. 

We also remind of the difference of the statement A=A expressing identity and B=A typically expressing that B inherits certain aspects of A (but not all). Often these aspects are not specified completely and the statement thus sometimes is open to confusion.  

The previous post discussed two different expressions of equality in physics: 

1. inertial mass = gravitational mass.
2. mass = energy, energy = mass 

Here 1. represents a truth resulting from viewing inertial mass to be an expression of (derived from) gravitational mass and thus the same. 

2. is Einstein's E=mc2, which is viewed to be the corner stone of modern physics. Since in physics energy is power to do work, it says that a body falling in a gravitational field doing work, such as the falling water in a Hydropower Plant, will loose energy (gravitational potential energy) and thus should loose mass according to 2. So there will be less water coming out than coming in to the plant, in contradiction to the physical principle of conservation of mass. A physicist would argue that the difference is too small to be noticed, but surely it is there, in principle. 

In any case a suspicious mind may wonder if that there is something fishy with the statement that energy = mass = energy (E=mc2). Ok, so more precisely Einstein does not say that mass and energy are equal, only that they are equivalent without specifying in what sense. But even so, as shown in the above example, a forced equivalence of mass and energy seems to lead into a strange contradiction.

It connects to Orwell's War = Peace which as identity is absurd, but as equivalence certainly has been used to justify a war by claiming that the aim is peace, that is Peace = War.  

We learn that equalities of the form B = A with B not identical equal to A, can (purposefully) be ambigious and need further specification to be meaningful. Seeking a specification of the meaning of energy = mass (E=mc2) may well lead to the conclusion that the equality does not make much sense. 

A contemporary physicist of course will claim that both chemical and nuclear exothermic reactions releasing energy suffers from a corresponding loss of mass according to E=mc2, which at least in the case of nuclear reactions is measurable and then shows to always exactly match E=mc2, nota bene if only properly measured. Any measurement not complying with a dogma of E=mc2 then must be corrected.

Phyisc is full of ambiguities as concerns equality. A first check is always to see if the units on both sides of the equality sign are equal. Sometimes this is also used to derive an equality by "dimensional" analysis.