studiot Posted November 30, 2021 Share Posted November 30, 2021 1 hour ago, Markus Hanke said: I didn’t mean to suggest that. Of course this needs to be done according to the proper rules and procedures of differential geometry in spacetime - the language of exterior calculus naturally lends itself to this. I remember MTW has a section that shows the proper procedure to construct volume integrals on semi-Riemannian manifolds; it’s that I was thinking of. A volume integral is not a volume. If you can assign a value of some property to every point in a particular region of space that property is either additive so that you get a total if you add up the values at every point. Such a property is called an extensive property. If the region is a line (1 dimensional) the taking a line integral gives that sum If the region is 2 dimensions then taking an area integral gives the sum If the region is 3 dimensional then a volume integral gives the sum If the region is 4 dimensional then taking a hypervolume integral gives the sum Examples would be mass, energy. If the property is not additive it is called an intensive property. Examples would be temperature, gas pressure. Intensive properties can be converted to extensive ones by taking deviations from an average. The average then represents every point in the whole. Link to comment Share on other sites More sharing options...

studiot Posted December 1, 2021 Share Posted December 1, 2021 8 hours ago, studiot said: If you can assign a value of some property to every point in a particular region of space Since 'fields' have been mentioned in connection with this subject It is perhaps worth noting that this is the Physics definition / usage of a field. It is not the Mathematics definition, which is somewhat different. If that property is a scalar then the field is a scalar field. If that property is a vector then the field is a vector field. If that property is a tensor then the field is a tensor field. The 'fundamental theorem of calculus' is the relation between the integral of that property over region and the integral over the boundary. A closed boundary will have one fewer dimensions than the region. Link to comment Share on other sites More sharing options...

Markus Hanke Posted December 2, 2021 Share Posted December 2, 2021 Are we not overthinking this a bit? An object such as the train mentioned in this thread is just a collection of (many) individual world lines. The geometric length of each individual world line between given events is something all observers agree on; thus, the volume implied by an entire bundle (congruence?) of such world lines should also be something everyone agrees on. Or am I seeing this wrong? Link to comment Share on other sites More sharing options...

md65536 Posted December 4, 2021 Share Posted December 4, 2021 (edited) On 12/1/2021 at 6:23 PM, Markus Hanke said: the volume implied by an entire bundle (congruence?) of such world lines should also be something everyone agrees on. A light-like path has a geometric length of zero, so all events on a given light cone should have a geometric (hyper)volume of zero. The interior of the cone would have positive volume (by choice of convention) or a time-like volume. The elsewhere would have negative or spacelike volume. The union of multiple light cones should have zero volume.* So for example if you have a light on the train, and turn it on at the beginning of the train's "life" and off at the end, then the geometric volume of all parts of the train that are lit over its entire life, is 0. If you consider only the lit part of the train (chop off everything outside the "light on" event's future light cone, and everything inside the "light off" event's future light cone), the train's entire existence has 0 geometric volume. Am I thinking about that reasonably? If so, I'm not sure if an invariant volume would make sense. It seems like a long-lived train, treated as a bunch of time-like world lines, should have a time-like volume. Or could you add up all the time-like world lines and get a different volume? * Or... did I make a mistake here? Maybe you can't just add up two light cones, because there are space-like intervals between events on the different cones, but then there are also time-like ones... Now I really don't see what a geometric volume could possibly mean. Edited December 4, 2021 by md65536 Link to comment Share on other sites More sharing options...

Markus Hanke Posted December 5, 2021 Share Posted December 5, 2021 5 hours ago, md65536 said: A light-like path has a geometric length of zero This is true only if you choose to parametrise the path using proper time - which physically just means that photons don’t have a rest frame. However, you can choose a different affine parameter, which will yield a non-zero result. Also, the world lines of the train’s constituent particles are time-like, so there shouldn’t be an issue. Link to comment Share on other sites More sharing options...

studiot Posted December 6, 2021 Share Posted December 6, 2021 On 12/4/2021 at 7:36 PM, md65536 said: A light-like path has a geometric length of zero, so all events on a given light cone should have a geometric (hyper)volume of zero. The interior of the cone would have positive volume (by choice of convention) or a time-like volume. The elsewhere would have negative or spacelike volume. The union of multiple light cones should have zero volume.* So for example if you have a light on the train, and turn it on at the beginning of the train's "life" and off at the end, then the geometric volume of all parts of the train that are lit over its entire life, is 0. If you consider only the lit part of the train (chop off everything outside the "light on" event's future light cone, and everything inside the "light off" event's future light cone), the train's entire existence has 0 geometric volume. Am I thinking about that reasonably? If so, I'm not sure if an invariant volume would make sense. It seems like a long-lived train, treated as a bunch of time-like world lines, should have a time-like volume. Or could you add up all the time-like world lines and get a different volume? * Or... did I make a mistake here? Maybe you can't just add up two light cones, because there are space-like intervals between events on the different cones, but then there are also time-like ones... Now I really don't see what a geometric volume could possibly mean. This comment might help you look further. Link to comment Share on other sites More sharing options...

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