4 ms·

What does it mean for a pair of events to be mutually distant?

raattgift 7y ago
Roughly, here I meant far enough away for the metric expansion of space to generate significant observables. If the cosmological redshift is not evident, then the pair of events are not "mutually distant" enough. In the "mutually distant enough" case, if the expansion is similar to that measured in our universe, then a RADAR pulse sent out by object A to object B would not return to object A before, say, about half of a sample of iron-60 at A had undergone beta decay (half-life 2.6 million years), all assuming that A and B are both moving slowly compared to the speed of light. The returning pulse will be at a significantly longer wavelength than the outgoing pulse. By contrast, a RADAR pulse that returns before half of a sample of carbon-11 has decayed by positron emission (half-life about 20 minutes), the returning pulse will be at pretty much exactly the same wavelength as the outgoing pulse. Here object A and object B move very slowly compared to the speed of light: the redshift is cosmological rather than special-relativistic. It works for neutrinos too, which have the advantage of always moving slower than the speed of light due to their small but nonzero invariant mass. In SI units, neutrino wavelengths vary from tiny fractions of a meter to several metres. We can measure these to an extent by looking for radiation from nuclear recoil reactions: shorter-wavelength means higher momentum and thus more and stronger recoil reactions. The less famous counterpart to the cosmic microwave background -- the https://en.wikipedia.org/wiki/Cosmic_neutrino_background https://en.wikipedia.org/wiki/Cosmic_neutrino_background -- is practically undetectable in this way. I say advantage because lightlike intervals are always zero by definition (that's why they're also called "null" intervals), so one has to use an affine parametrization of the interval to or otherwise fix coordinates and units to compare how far apart in spacetime events connected by RADAR signals are. The intervals of events connected by neutrino beams ("nadar?") are timelike, and so we can more straightforwardly consider the contribution of the cosmological constant to the (nonzero) magnitude of the interval \Delta s^2. But neutrinos are still ultra-relativistic -- simultaneously emitted neutrinos and photons (say, from extragalactic supernovae) are detected practically simultaneously by instruments on and around Earth. (In practice, such simultaneously-emitted neutrinos can win races to our detectors because the universe is generally more transparent to them than to the photons emitted from the same event: the relative opacity slows down the latter).

Roughly, here I meant far enough away for the metric expansion of space to generate significant observables. If the cosmological redshift is not evident, then the pair of events are not "mutually distant" enough. In the "mutually distant enough" case, if the expansion is similar to that measured in our universe, then a RADAR pulse sent out by object A to object B would not return to object A before, say, about half of a sample of iron-60 at A had undergone beta decay (half-life 2.6 million years), all assuming that A and B are both moving slowly compared to the speed of light. The returning pulse will be at a significantly longer wavelength than the outgoing pulse. By contrast, a RADAR pulse that returns before half of a sample of carbon-11 has decayed by positron emission (half-life about 20 minutes), the returning pulse will be at pretty much exactly the same wavelength as the outgoing pulse. Here object A and object B move very slowly compared to the speed of light: the redshift is cosmological rather than special-relativistic. It works for neutrinos too, which have the advantage of always moving slower than the speed of light due to their small but nonzero invariant mass. In SI units, neutrino wavelengths vary from tiny fractions of a meter to several metres. We can measure these to an extent by looking for radiation from nuclear recoil reactions: shorter-wavelength means higher momentum and thus more and stronger recoil reactions. The less famous counterpart to the cosmic microwave background -- the https://en.wikipedia.org/wiki/Cosmic_neutrino_background https://en.wikipedia.org/wiki/Cosmic_neutrino_background -- is practically undetectable in this way. I say advantage because lightlike intervals are always zero by definition (that's why they're also called "null" intervals), so one has to use an affine parametrization of the interval to or otherwise fix coordinates and units to compare how far apart in spacetime events connected by RADAR signals are. The intervals of events connected by neutrino beams ("nadar?") are timelike, and so we can more straightforwardly consider the contribution of the cosmological constant to the (nonzero) magnitude of the interval \Delta s^2. But neutrinos are still ultra-relativistic -- simultaneously emitted neutrinos and photons (say, from extragalactic supernovae) are detected practically simultaneously by instruments on and around Earth. (In practice, such simultaneously-emitted neutrinos can win races to our detectors because the universe is generally more transparent to them than to the photons emitted from the same event: the relative opacity slows down the latter).