In the book Relativity: The Special and General Theory by Einstein I have came cross the following sentence and here I quote "Hereupon we introduce a hypothesis: that the influence of the gravitational field on measuring rods, clocks and freely−moving material points continues to take place according to the same laws, even in the case where the prevailing gravitational field is not derivable from the Galilean special care, simply by means of a transformation of co−ordinates." I don't understand what Einstein exactly meant when he said "even in the case where the prevailing gravitational field is not derivable from the Galilean special care" what is a gravitational field which is derivable from the Galilean special case? And how it looks like that gravitational field which isn't derivable from this Galilean spacial case? I get that he originally pictured gravity as being produced when things are interpreted properly in an accelerated frame, but not every gravitational field can produced in such a way, so could it be that this kind of a gravitational field, that one produced when working with accelerated frame, the kind Einstein referred to as Galilean derivable? And as a final question, What kind of coordinate transformation did Einstein exactly mean?
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By a gravitational field that is "derivable from the Galilean special case" I think that he means the field experienced in a non-inertial frame such as an elevator acceleraing upwards. This is obtained by a coordinate transformation $x\to x'= x+ (1/2)a t^2$. Such a gravitational field will have no tidal forces.
The usual assumption in the equivalence priciple is that, even in the presence of tidal forces where we cannot establish a global inertial frame, in a small enough region you can ignore the tidal forces and use the usual equations of special relativity.
answered Dec 21, 2022 at 15:16