I'm trying to learn something from Chapter 3 of Mach's book, but he's showing how the principles of mechanics are used, which means he is mostly deriving results that I mostly already know, and via methods that are just close enough to the derivations that I've seen that it's hard to get the "Huh, that's different" feeling that makes me want to read a derivation of something I already know.
Hopefully this will pick up. Otherwise I'll be skimming the next 200 pages.
Friday, August 14, 2015
Monday, August 10, 2015
Bucket list
I'm reading Mach's discussion of the infamous "bucket problem." I feel like Mach never would have written this part, and doomed himself to so much criticism in later years, if he had written it before his discussion of absolute time.
He starts off by critiquing Newton's idea of time. Newton tried to define absolute time in the Principia, with a definition that could be summed up as "Absolute time is, you know, time." He then defines relative time as a measurement of absolute time, a measurement made by observing the motion of bodies. Mach calls him on that, and appropriately so. We measure time by the motion of the earth, or the motion of a pendulum, or the oscillations of a circuit or vibrations of a crystal, but how do we know that the events we are marking off are "really" evenly separated in time? The short answer is that we don't know if they're "really" separated by equal time intervals, but when we construct theories on that assumption we get a picture that is self-consistent and also consistent with measurement. Mach pushes on that a bit, and notes that all time is relative to some clock, and whether or not we measure the times between a series of events to be steady depends on whether they match up with the ticks of a clock, not whether they're "really" the same. (You might say something about our body's internal clocks, but how do we know that heartbeats or other physiological events are "really" steady?)
So far, so good. If he had pushed a bit farther on this, and considered the right thought experiment, he might even have constructed the Special Theory of Relativity (at which point he assuredly would have pointed out conceptual inadequacies in its construction, while also citing numerous historical precursors, because that's how Mach rolls.)
Even better, he notes that all we can really do is compare perceptions and memories, and goes so far as to notice the Thermodynamic Arrow of Time: If we have a memory of two objects at different temperatures being placed in contact and then left undisturbed, later we will have a perception of a smaller temperature difference.)
But then Mach goes on to make a point about how distances are measured with respect to objects taken as fixed for the sake of argument (true), and also argues that we can never really know what would happen in some case that we haven't measured. From there, he leaps to Newton's bucket: Newton argued that we know that a bucket is spinning because if makes the water rise up along the sides, whereas if we run around the bucket the water does not rise. Newton, who stood on the shoulders of Galileo, saw this as a fundamental difference between inertial and non-inertial frames (even though that didn't stop him from attempting definitions of "absolute space" and "absolute motion"). Mach says that since everything is measured with respect to something else, we don't really know what would happen if we set the entire earth in motion at the same rate as the bucket because we can't manipulate an object that large. It's confusing.
I feel like Mach could have been Einstein with a bit of luck and maybe a bit more of the right kind of boldness and willingness to accept some inadequacies in reasoning. Einstein went pretty far by asking "Well, what does it really mean to say that something is moving with respect to something else?" and "Well, what does it really mean that gravitational fields and accelerated frames give rise to the same types of phenomena?" That's not so different from Mach's program in this book, except that Einstein was OK with making leaps while Mach was pointing out how inadequate concepts are.
He starts off by critiquing Newton's idea of time. Newton tried to define absolute time in the Principia, with a definition that could be summed up as "Absolute time is, you know, time." He then defines relative time as a measurement of absolute time, a measurement made by observing the motion of bodies. Mach calls him on that, and appropriately so. We measure time by the motion of the earth, or the motion of a pendulum, or the oscillations of a circuit or vibrations of a crystal, but how do we know that the events we are marking off are "really" evenly separated in time? The short answer is that we don't know if they're "really" separated by equal time intervals, but when we construct theories on that assumption we get a picture that is self-consistent and also consistent with measurement. Mach pushes on that a bit, and notes that all time is relative to some clock, and whether or not we measure the times between a series of events to be steady depends on whether they match up with the ticks of a clock, not whether they're "really" the same. (You might say something about our body's internal clocks, but how do we know that heartbeats or other physiological events are "really" steady?)
So far, so good. If he had pushed a bit farther on this, and considered the right thought experiment, he might even have constructed the Special Theory of Relativity (at which point he assuredly would have pointed out conceptual inadequacies in its construction, while also citing numerous historical precursors, because that's how Mach rolls.)
Even better, he notes that all we can really do is compare perceptions and memories, and goes so far as to notice the Thermodynamic Arrow of Time: If we have a memory of two objects at different temperatures being placed in contact and then left undisturbed, later we will have a perception of a smaller temperature difference.)
But then Mach goes on to make a point about how distances are measured with respect to objects taken as fixed for the sake of argument (true), and also argues that we can never really know what would happen in some case that we haven't measured. From there, he leaps to Newton's bucket: Newton argued that we know that a bucket is spinning because if makes the water rise up along the sides, whereas if we run around the bucket the water does not rise. Newton, who stood on the shoulders of Galileo, saw this as a fundamental difference between inertial and non-inertial frames (even though that didn't stop him from attempting definitions of "absolute space" and "absolute motion"). Mach says that since everything is measured with respect to something else, we don't really know what would happen if we set the entire earth in motion at the same rate as the bucket because we can't manipulate an object that large. It's confusing.
I feel like Mach could have been Einstein with a bit of luck and maybe a bit more of the right kind of boldness and willingness to accept some inadequacies in reasoning. Einstein went pretty far by asking "Well, what does it really mean to say that something is moving with respect to something else?" and "Well, what does it really mean that gravitational fields and accelerated frames give rise to the same types of phenomena?" That's not so different from Mach's program in this book, except that Einstein was OK with making leaps while Mach was pointing out how inadequate concepts are.
Sunday, August 9, 2015
Mach-ing waves
I have never before read a physics book that included a three page excerpt from a history of Alexander the Great's expedition in India. But Mach does exactly that in chapter 2, section IV, part 7. He is discussing tides, and notes that Alexander's armies, being used to the minimal tides of the Mediterranean, were wholly unprepared for the Indian ocean.
It is important to be reminded that people named Alex are of great ambition. It is strange how few great physicists are named Alexander, Alexis, Aleksandr, Oleksandr, Alejandro, Alessandro, etc., but at least we Alexes can claim Alexander Fleming in our ranks.
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