Symmetry of “Inertial Mass”

In any elastic collision between high velocity particles, conservation of momentum, expressed in its naive form in terms of the initial frame of either particle, requires that the “inertial” mass used for the other (and for both of the post-collision particles) include the relativistic gamma factor(s). So it is indeed true that twins moving at constant velocities in different directions BOTH infer the other’s inertial mass to be more than their own.

Source: (1000) Alan Cooper’s answer to Ok, Newton’s physics was wrong. But is there any experimental evidence suggesting Einstein’s relativity claim is true that twins moving at constant velocities in different directions BOTH measure the other’s inertial mass to be more than his own? – Quora

What is relative in Relativity?

What is relative in any physical theory of “relativity” are the space-time coordinates of events from the perspectives of different observers. 

One problem, I think, with the names using ‘Theory of Relativity’ is that they seem to suggest theories about what is relative, rather than (more correctly) about how the coordinates used by different observers need to be related in order to ensure that the laws of physics are invariant (ie NOT relative).

In fact the coordinates that seem most natural to any observers for the purpose of expressing their experience in quantitative terms are always to some extent relative to the observers, so just saying that they are relative without specifying how is not telling us much (though in the new theories there is “more” relativity in the sense that time as well as the spatial coordinates becomes relative).

Our intuitively expected relationship between the coordinates of relatively moving observers allows all observers to use the same time coordinate, and so to agree on which events are simultaneous (ie constitute the same moment in time). It also preserves the form of Newton’s equations of motion for observers moving at constant relative velocity – which, as Galileo noted, has the consequence that observers moving with constant relative velocities cannot, by mechanical experiments, identify any particular one as being stationary. So the question of who is moving can only be answered relative to a particular observer – but this is just one particular instance of the relativity of coordinates.

[Sometimes observers moving relative to some larger object such as the Earth might choose to agree on a fixed Origin based on that object rather than on their own positions. But Galileo noted that if they are all moving together inside a moving vessel without any view of the outside, then it makes sense for them to use the vessel itself as their frame of reference – and relative to that, anything outside would appear to be moving in the opposite direction. In the world of Galilean/Newtonian physics there is nothing aside from its greater size which makes us prefer the Earth’s frame to that of the vessel, nor anything besides Earth’s proximity which makes us prefer its frame to that of the Sun. The answer to whether or not anything is or is not actually moving was thus, even in classical mechanics, entirely relative to the observer’s arbitrary choice of a frame of reference; and so that certainly was NOT anything new in Einstein’s theory.]

The above noted preservation of form of the equations of motion is perhaps confusingly called both “Galilean invariance” and “Galilean relativity”. The confusion could be avoided by making it clear that the word “relativity” applies to coordinates and “invariance” to the laws of physics. But I think that the practice of using “relativity” for the invariance itself rather than for the coordinate transformations under which it holds was indeed a misnomer which I believe precedes Einstein (though as an aside I must add that it seems surprisingly difficult to find out who was actually the first to do this).

Einstein’s special theory describes how the spacetime coordinates must be related in order for the laws of electromagnetism to have the same form for all inertial (ie unaccelerated) observers in the absence of any gravitational field. It turns out that for this to work, observers in relative motion will not be able to use the same time coordinates, and indeed will have different notions of simultaneity; so in this theory there is indeed something more that is “relative” than in the Galilean theory (but I don’t think that is why the theory got its name).

Einstein’s theory derives the relativity of simultaneity, and the formulas relating spacetime coordinates of different observers, from the principle of invariance of Maxwell’s equations (and so in particular, invariance of the speed of light) from the points of view of all inertial observers. But in my opinion Einstein’s reference to that principle as the “principle of relativity” (as opposed to the “principle of invariance” as suggested for example by Felix Klein) was indeed a misnomer, and apparently even Einstein eventually expressed some agreement with this  (but too late to actually change it).

[The special theory of relativity also includes modifications of the laws of mechanics (excluding gravity) which are necessary for them to remain invariant under the same transformations as those which preserve Maxwell’s equations – but this has nothing to do with the name except for the fact that perhaps the thinking was that the “principle” in question was that all physical laws need to be invariant under the same relativity of coordinates.]

The general theory goes on beyond the special theory to describe how the coordinates should be related in order to preserve an invariant form for both electromagnetic and gravitational forces under more general conditions (including accelerated observers and gravitational fields). So it’s not that more things are relative in the general theory, but rather that the relativity of the same things is explored under a more general range of conditions.

P.S. It should perhaps be noted that, just as the special theory has no distinguished inertial frame, the general theory does not provide any purely local way to distinguish inertial from accelerated frames as no accelerated observer can distinguish the experience of being accelerated from that of being prevented from falling freely in some “fictitious” gravitational field – which can only be identified as truly fictitious by observing the absence of possible sources (mass-energy distributions) out to an arbitrarily great distance. So there is some sense in which acceleration vs gravitation distinction is not quite absolute in the general theory but I don’t think that this (or the absence of any distinguished inertial frame in the special theory) was ever the reason for our use of the word “relativity”.

Source: I am trying to understand the term general relativity: what is relative in GR? Gravitation (acceleration) is absolute, not relative! So what is relative in GR? Is it perhaps a misnomer? – Quora