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## Main Question or Discussion Point

Blandford & Thorne,

Blandford & Thorne's clocks and rulers are regarded as in some sense "ideal", yet not ideally accurate. What quality does their idealness consist of if not accuracy? (I take it it's not that they're just a very nice colour and you can check your emails on them.) Is it just that that there's a scale limit on their accuracy, so that, given a finite degree of accuracy of instruments, you can always choose smaller and smaller scales till you find a scale where they detect no curvature--rather than the empty statement that given a degree of curvature, you can always find instruments not accurate enough to detect it!

I notice that Blandford and Thorne only specify a spatial accuracy at this stage; is that significant?

Besides curvature, could topology limit the size of a Lorentz frame of a given accuracy?

The metric tensor field is defined at each point. Its value at each point contains information about curvature. Sometimes the value is derived by an infinitesimal analogue of the Pythagorean formula. If curvature is significant enough that it can't be neglected in such a small region as one point, how can it be neglected in such a large region as a space shuttle? Is it because different degrees of accuracy are being used in these two cases, i.e. this way of talking about the metric tensor field assumes ideal, unlimited accuracy (and if so, how does that mesh with the idea of a differential a linear approximation)?

*Applications of Classical Physics*:Taylor & Wheeler,One of Einstein's greatest insights was to recognize that special relativity is valid not globally, but only locally, insidelocally freely falling (inertial) reference frames. Figure 23.1 shows a specific example of a local inertial frame: The interior of a space shuttle in earth orbit, where an astronaut has set up a freely falling (from his viewpoint “freely floating”) latticework of rods and clocks. This latticework is constructed by all the rules appropriate to a special relativistic,inertial (Lorentz) reference frame[...] However, there is one crucial change from special relativity: The latticework must be small enough that one can neglect the effects of inhomogeneities of gravity (which general relativity will associate with spacetime curvature; and which, for example, would cause two freely floating particles, one nearer the earth than the other, to gradually move apart even though initially they are at rest with respect to each other). The necessity for smallness is embodied in the word “local” of “local inertial frame”, and we shall quantify it with ever greater precision as we move on through this chapter. [...] We shall use the phraseslocal Lorentz frameandlocal inertial frameinterchangeably

*Spacetime Physics*:These definitions seem to be based on the notion of a "physical" or "practical" infinitesimal: a quantity too small to be detected. But how can we measure the accuracy of an imaginary detector? Taylor & Wheeler answer this by saying: you decide. In that case, could we not define a (trivial) global Lorentz frame if we specify zero accuracy, and a Lorentz frame of any other size, from infinite down, by an appropriate choice of accuracy? This seems at odds with the connotation of smallness.A reference frame is said to beinertialin a certain region of space and time when, throughout that region of spacetime, andwithin some specified accuracy, every test particle that is initially at rest remains at rest, and every test particle that is initially in motion continues that motion without change in speed or in direction. An inertial reference frame is also called aLorentz reference frame. In terms of this definition, inertial frames are necessarily alwayslocalones, that is inertial in a limited region of spacetime.

Blandford & Thorne's clocks and rulers are regarded as in some sense "ideal", yet not ideally accurate. What quality does their idealness consist of if not accuracy? (I take it it's not that they're just a very nice colour and you can check your emails on them.) Is it just that that there's a scale limit on their accuracy, so that, given a finite degree of accuracy of instruments, you can always choose smaller and smaller scales till you find a scale where they detect no curvature--rather than the empty statement that given a degree of curvature, you can always find instruments not accurate enough to detect it!

I notice that Blandford and Thorne only specify a spatial accuracy at this stage; is that significant?

Besides curvature, could topology limit the size of a Lorentz frame of a given accuracy?

The metric tensor field is defined at each point. Its value at each point contains information about curvature. Sometimes the value is derived by an infinitesimal analogue of the Pythagorean formula. If curvature is significant enough that it can't be neglected in such a small region as one point, how can it be neglected in such a large region as a space shuttle? Is it because different degrees of accuracy are being used in these two cases, i.e. this way of talking about the metric tensor field assumes ideal, unlimited accuracy (and if so, how does that mesh with the idea of a differential a linear approximation)?