Let us illustrate this with only a few quotations:They end up settling on the definition used by Lorentz in 1895, and copied by Einstein in 1905. But then they note that it does not match the modern definition:
“the laws of physical phenomena should be the same, whether for an observer fixed, or for an observer carried along in a uniform movement of translation” (Poincaré 1956, p. 167);
“If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K0 moving in uniform translation relatively to K.” (Einstein 1923. p. 111);
“it is impossible to measure or detect the unaccelerated translatory motion of a system through free space or through any ether-like medium” (Tolman 1949, p. 12);
“all physical phenomena should have the same course of development in all system of inertia, and observers installed in different systems of inertia should thus as a result of their experiments arrive at the establishment of the same laws of nature” (Møller 1955, p. 4);
“the laws of Physics take the same mathematical form in all inertial frames” (Sardesai 2004, p. 1);
“The same laws of nature are true for all inertial observers.” (Madarász 2002, p. 84)
“The uniform translatory motion of any system can not be detected by an observer traveling with the system and making observations on it alone.” (Comstock 1909, p. 767);
“The laws of nature and the results of all experiments performed in a given frame of reference are independent of the translational motion of the system as a whole. More precisely, there exists a [...] set of equivalent Euclidean reference frames [...] in which all physical phenomena occur in an identical manner. (Jackson 1999, p. 517);
“If we express some law of physics using the quantities of one inertial frame of reference, the resulting statement of the law will be exactly the same in any other inertial frame of reference. [...] we write down exactly the same sentence to express the law in each inertial frame.” (Norton 2013);
“all inertial frames are equivalent for the performance of all physical experiments” (Rindler 2006, p. 12);
“the laws of physics are invariant under a change of inertial coordinate system” (Ibid., p. 40);
“The outcome of any physical experiment is the same when performed with identical initial conditions relative to any inertial coordinate system.” (Ibid.);
“experience teaches us that [...] all laws of physical nature which have been formulated with reference to a definite coordinate system are valid, in precisely the same form, when referred to another co-ordinate system which is in uniform rectilinear motion with respect to the first. [...] All physical events take place in any system in just the same way, whether the system is at rest or whether it is moving uniformly and rectilinearly.” (Schlick 1920, p. 10);
“laws must be Lorentz covariant. Lorentz covariance became synonymous with satisfaction of the principle of relativity” (Norton 1993, p. 796);
“The laws of physics don’t change, even for objects moving in inertial (constant speed) frames of reference.” (Zimmerman Jones and Robbins 2009, p. 84);
“the basic physical laws are the invariant relationships, the same for all observers” (Bohm 1996, p. viii);
“laws of physics must satisfy the requirement of being relationships of the same form, in every frame of reference” (Ibid. p. 54).
RP and the covariance of equations E are not equivalent — in contrast to what is so often claimed in the literature. As Norton (1993, p. 796) writes:The term "relativity principle" and its popularization is from Poincare and his 1902 book. So his definition ought to be the controlling one. He then proved Lorentz covariance in 1905, and Minkowski used that as the basis of his 1908 spacetime theory. After that, everyone used covariance, and not the weaker Lorentz-Einstein condition.The lesson of Einsteins’s 1905 paper was simple and clear. To construct a physical theory that satisfied the principle of relativity of inertial motion, it was sufficient to ensure that it had a particular formal property: its laws must be Lorentz covariant. Lorentz covariance became synonymous with satisfaction of the principle of relativity of inertial motion and the whole theory itself, as Einstein (1940, p. 329) later declared:The content of the restricted relativity theory can accordingly be summarized in one sentence: all natural laws must be so conditioned that they are covariant with respect to Lorentz transformations.
Norton is a little misleading, because Einstein's 1905 paper said nothing about covariance, and only about the weaker Lorentz 1895 notion. It is true that Lorentz covariance became synonymous with satisfaction of the principle of relativity, but that is because Poincare proved it in 1905 and Minkowski popularized it in 1908.
The Lorentz principle, used by Einstein, was that the equations have the same form in different frames. Covariance means that the equations have a unified geometric meaning that automatically subsumes the equations for the different frames.
Strictly speaking, covariance is a mathematical principle and RP is a physical principle. Poincare wrote about the relativity of space, meaning that you can just measure distances relative to other points, and you cannot deduce an absolute coordinate for position in space. Likewise, the relativity of velocity says you cannot measure your absolute velocity. Covariance becomes a physical principle after variable are identified with physically measurable entities.
Einstein's 1905 relativity paper is one of the most famous science papers every written, and yet people are still getting it wrong a century later. As quoted above, he said "the same laws hold good". That is, the law in one frame has the same mathematical form as the law in another frame. Poincare made the superior statement that the laws "should be the same". The laws do not just look the same, they are the same. Einstein did not appreciate how covariance makes this stronger statement possible.
This is all detailed in my book, How Einstein Ruined Physics.