B "Spacetime Physics" by Taylor and Wheeler

[whatif]

I am also confused about why this experiment can't be done for more than 8 seconds unless ofcourse the reason is that the train will have crashed into Earth in 8 seconds.

This is the way I think about it. I think it is less abstract and I hope it is correct.

For all three directions- and for all intermediate directions- let it be found by calculation that the relative drift of two test particles equals half the minimum detectable amount or less.

The required accuracy is that the measuring instruments are not able to detect the effects of gravity. Taylor and Wheeler are saying that at the particular location (relative to the Earth's surface) and for the required accuracy, the experiment must be performed within the spatial dimensions and within the time dimension, without explaining how they know and without reference to the dimensions (however, that is not clear from the way they express it).

Then throughout a cube of space 20 meters on an edge and for a lapse of time of 8 seconds, (which is how long it would take the train to hit the ground of the Earth I think), 2400 million meters of light travel time being 8 seconds, test particles moving every which way depart from straight line motion by undetectable amounts.

What is meant is that for those spatial dimensions the experiment must be done within 8 seconds to achieve the required accuracy, without explaining how they know. Any longer than 8 seconds and gravitational effects would be detectable, meaning the region of spacetime could not be treated as a free-float frame (this is clear). It so happens that the train will crash into the Earth just after 8 seconds but that is not a consideration as to whether the region of spacetime can be treated as a free float frame.

I am really clueless about what he means when he says: "let it be found by calculation that the relative drift of two test particles equals half the minimum detectable amount or less"??

Taylor and Wheeler are just telling the reader that region of spacetime meets the requirements of accuracy but they have expressed themselves clumsily.
They are not explaining how they arrived at that knowledge. They are only trying to explain that the dimensions of regions of spacetime being treated as free float frames are limited by the required accuracy (based on the measuring instruments not being able to detect the effects of gravity).

More interesting for me though is the calculation of the region of spacetime and it's dimensions since it seems to describe the nature of spacetime. What exactly is this 20 by 20 by 20 by 2400 million meter "region" he is referring to?

I do not think Taylor and Wheeler intended for the result of the product to be calculated. It was just stating the dimensions. Taylor and Wheeler do not make any use of the calculated product. As per Mister T:

Consider a cube measuring 20 m by 20 m by 20 m. Study its contents for 8 seconds.

8 seconds being the time that region of space can be treated as a free float frame to meet the required accuracy because Taylor and Wheeler told us so (effectively telling the reader that the effects of gravity will be undetectable by the instruments used in that region of space time).

Multiplying time and space would make little sense in Newtonian physics (as normally constructed, anyway - I won't guess about Cartan's geometric reformulation), because time is something completely different from space. It makes perfect sense in a 4d spacetime, however, where time and space are the same kind of thing.

These kinds of statements apply to the mathematics (geometric units being used). They are abstract statements. It allows, for example, for time to be measured in meters by use of the speed of light constant, c. However, a meter of distance is not the same as a meter of time. I have no problem with the mathematics, because it simplifies the analysis. However, these kinds of statements and analogies used to explain them are misleading, in my opinion. A volume of spacetime, being a calculation based on 3 dimensions of space and one dimension of time is a mathematical number with units. However, one needs to remain aware that the same volume can represent two different things, space and time, in different proportions (to say space and time are the same kind of thing is misleading ,in my opinion).

 

[Ibix]

However, one needs to remain aware that the same volume can represent two different things, space and time, in different proportions

If you are going to say this, you should also say that one needs to remain aware that the exact same region in spacetime (not the volume number but the geometric figure, the 20x20x20x(2.4x10⁹) hypercuboid in this case) is itself different amounts of space and time, because the foliation of spacetime into space and time is arbitrary and can be done in different ways.

 

[PeterDonis]

You get around the crashing concern by imagining a tunnel bored through the center of the planet large enough for the train to pass through freely. The experiment can be conducted for any length of time.

No, it can't. The train can indeed execute simple harmonic motion passing through the tunnel, but tidal gravity will affect the marbles inside it. The point of the 8 seconds is that, given the limits of detectability of tidal gravity that the textbook is assuming, that's how long you can let the train fall freely without detecting tidal gravity effects.