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I am new to relativity, but I have some familiarity with differential geometry.

If I ask a mathematician what the difference is between a change of coordinates (or basis) on a vector space and an operator on the space that does the inverse, he will say that the effect is the same.

Let's limit ourselves for now to think of active rotations of $\mathbb{R}^2$, the underlying space rotating, versus a passive rotation, a change of orthonormal basis on $\mathbb{R}^2$, the underlying space remaining fixed.

A mechanician says there is definitely a difference between a change of coordinates, or reference frame, and the active rotation. An active rotation will induce centripetal accelerations in a continuum being rotated, and stresses will develop. On the other hand, an observer rotating in a non-inertial reference frame observes the continuum as accelerating, but doesn't measure any stress. Therefore, he knows that he is making observations from a non-inertial frame.

I am new to the concept of boost. I would like to understand its analogue to the preceding viewpoint. Suppose we are in a 1D space manifold. Then spacetime is 2D. Two observers don't move and are staring at the same train. One observer gets boosted. He observes the train under compression (length contraction) despite no apparent forces acting to compress the train. Therefore, he understand that he is being boosted and is making observations from a boosted frame. If he knows his history, then once he has reached constant velocity, he will be able to account for the change in train length as measured by both observers and reconcile their difference.

We arrange the coordinate system of the observer who remains fixed relative to the train as a right handed system in (his?) spacetime. Initially, both observers have coincident coordinate systems. Then the (instantaneous) boost happens. I can either think of the effect on the boosted observer as a change of coordinates from the fixed observer under a hyperbolic rotation, with the underlying spacetime manifold remaining fixed, OR I can imagine the coordinate system remaining perpendicular, and the underlying spacetime experiencing a pure shear deformation.

Which is the better conception and why? I am tempted to see the spacetime manifold as an invariant, with all effects of relativity being described in terms of a change of coordinates. But I am unsure about the process of fixing spacetime once and for all, as drawing it, for example, as a square of events in $\mathbb{R}^2$. After all, isn't everyone's coordinate system orthogonal for "their" spacetime?

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  • $\begingroup$ There are some misconceptions involved in this question. There is no such thing as a boost in general relativity, and a coordinate system is not the same as a frame of reference. GR doesn't have global frames of reference. See physics.stackexchange.com/questions/458854/… $\endgroup$
    – user4552
    Aug 1, 2019 at 23:56
  • $\begingroup$ @BenCrowell: There is no such thing as a boost in general relativity This is sloppy and untrue statement. Boosts are elements of Lorentz group that have a canonical global action defined on Minkowski space. Minkowski space is a solution of GR, so there are boosts (defined globally) in general relativity. $\endgroup$
    – A.V.S.
    Aug 2, 2019 at 14:12

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A boost is a particular change of coordinates in Minkowski spacetime and nothing more. Thinking of it as any kind of deformation of spacetime is a terrible misconception, in my opinion. To me, any “deformation” would have to change the spacetime interval between two events, and boosts do not do this.

It is also a misconception to think that there is a preferred or fixed set of coordinates. All inertial frames related by boosts (or, more generally, Poincaré transformations) are completely equivalent.

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