Killtech

Thanks. I understand that construction. Note that it is called the "geodesic clock" and as the name suggests it is constructed via geodesics. A geodesic is defined via the geometry and more precisely the Christoffel symbols derived from the metric tensor of said geometry. As i tried to highlight, the definition metric tensor already implies a what clocks and length measurements must produce. The theory of relativity never formally defines what we should count as a clock even though Einstein uses the term a lot. But we can't just use any oscillators because each may be impacted differently by the environment at a given location and frame (unless we properly correct for the deviances). The geodesic clock is actually a theoretic derivation what clocks we must use to be to consistent with Einsteins theory and atomic clocks are constructed/defined and corrected in just such a way to be as close as possible to that ideal. This is why what you posted is so very significant - it gives us the final details for a most accurate interpretation of the theory. I am however looking at using a time standard instead of proper time which is a major difference. I should maybe again highlight that TDB is normally not thought in terms of clocks for the good reason to avoid confusion. In fact TDB time has to be calculated via an atomic clock (which is the most accurate implementation of an ideal clock of general relativity) and correcting it by the influence of gravity (which requires additional information) so it is not just more complex method, but also explicitly deviates from what a theoretical clock should yield. Note that most clocks we use in daily life are invalid in the context of relativity because they don't measure proper time but also give time according to a time standard. So if you were to climb Mount Everest with two clocks, one radio-synched watch and an atomic clock we will find a discrepancy: the UTC time standard simply ignores our actual location whereas an atomic clock doesn't, hence general relativity will make them run apart over time - yet neither is going off. They just measure different things! It is important to understand that. What i am looking for is the question if physics skipped the part where the mathematical discipline of Riemann geometry is so flexible that it allows us to treat TDB as if it was a time measure we could base clocks on for physical experiments. Of course it has practically a limited use since measuring time in SI-seconds allows for a much greater accuracy the measurement in TDB which inherit a big uncertainty from the gravitational correction which is hard to measure and calculate precisely. Then again, if we want to calculate whether two celestial bodies will eventually collide, we have that error one way or another.

This is the bit what i am currently struggling with in physics. Relativity is a concept deeply rooted in the nature of geometry itself because we must always account that space and time are never measured absolutely but in units - and those are always relative to some local and frame dependent reference. For the SI second this is a specific Caesium transition frequency at the observers frame and location. When something is said to be constant it merely means it changes exactly the same as the reference in units of which it is measured. Because all measurement is relative, it is impossible/meaningless to determine if something changes in an absolute sense, hence it is crucial to understand the entire geometric formulation is about relative metrics and distances. But from what i learn from physics, i did get the impression it lacks understanding of the full extend of relativity and its seems stuck in interpreting everything within a singular view. So the baryocentric metric representation discussed here is just one example to show how relative the entire geometry is beyond what is usually used in physics. To go into detail, let's start with a motivation. In the beginning of the 19th century, time didn't not have the same importance it has in our todays society. Back then each town might have had its own non uniform local time and clocks. Only when rail tracks between the cities were build, time became of higher importance as we needed to organize events in between different locations. The railway time was introduced as an unifying standard and one of the early time standards which effectively became GMT/UTC our clocks still reference today. In general relativity we have a little bit of a similar scenario with each frame having its very own proper time and clock. The reasons may differ, but the practical problem this creates are the same, hence IAU introduced the TDB coordinate time [https://en.wikipedia.org/wiki/Barycentric_Dynamical_Time] that serves the purpose of 'railway time' for the solar system. Let's assume i would be an interplanetary traveler and have to buy a clock, then what concept of time would i want it to show? or let's say i want to record the historic accounts of a interstellar civilization - what time should i use? Proper and coordinate times are very different representations of time for different purposes, both perfectly valid in their own right. But for our intuition, the one is a lot more abstract then the other, so i was wondering what happens when we formulate general relativity in a different time? Measuring time differently, or rather measuring a different time has a lot of implications. TDB corrects for the effects of gravity fields to proper time - so at the edge of the solar system SI and TDB clocks may yield same numerical results, but on earth TDB clocks will run relatively faster. Consequently, if we were to use such clocks to measure the speed of light in vacuum, we find that light signals appear to travel slower in earths gravity well then at the edge of the solar system. There is no contradiction here because TDB clocks results have to use an own coordinate time unit (i.e. not the SI second), and these units are locally different from SI, allowing no simple comparison. Mathematically however, we do have the means to adequately predict how physics will look when we keep using such units. Now this is where the discipline of geometry comes into play. First, let's recount what is mathematically the difference between a sphere and an ellipsoid Riemann manifolds? Technically we can take the very same subset of R^3 for both manifolds and use the very same coordinates for either. The actual difference is in fact how we define to *measure* length along a line element. Or more generally, the metric tensor we define on the manifold. Even though geometry is a mathematical discipline the practical concept of measurement is a fundamental part of the theory (as the term "metric" implies) - which the metric tensor represents in a abstract way. Now going in back to TDB clock question, we need a metric tensor that is correctly able to represent TDB time along a line element. We can do that do that with length as well i.e. using BCRS coordinate lengths [https://en.wikipedia.org/wiki/Barycentric_and_geocentric_celestial_reference_systems] to yield a new metric tensor representing measurements done with the corresponding coordinate units. Admittedly it does not make practical sense with just any coordinates, but for these, the results can be well interpreted. For example the coordinate distance between earth and moon is almost constant since it does not depend on the overserves frame i.e. it is always a specific proper distance. Geometrically speaking, we exchanged the metric tensor g of general relativity for another - and i don't mean its coordinate representation - I really mean to have exchanged the whole geometry. Because we exactly know how these two metric tensors relate to each other at every location, we can write down a transformation procedure to reformulate Einsteins field equations with their original metric tensor to the geometry implied by the TDB-BCRS metric tensor. Of course that does not change physics in any way - we just change the point of view (i.e. measurement) to yield a different model of physics but with a new interpretation (mapping time and space to different clocks and measurements). If we consistently exchange a physical model along with its interpretation for another, the changes in both cancel each other out in a sense: mathematically speaking we have a commuting diagram [https://en.wikipedia.org/wiki/Commutative_diagram] where the interpretations takes the role of the morphisms while the models and reality are the nodes/vertices. If we were to look at Maxwells equations in barycentric coordinate units specifically, the transformation prescribed from Einsteins metric tensor to the barycentric geometry will introduce a gravity induced refractive index to the vacuum, stemming from the gravity correction to TDB clocks. As such in the new model (and its units) the speed of light is not a constant and the bending of light by gravity is now described via refractive index instead of the geometry. The actuary predictions however remain ultimately identical. I should also mention that the barycentric nature of these coordinates means the new model has chooses a preferred frame. and since it measures time intervals and length in coordinates differences i.e. entirely invariant of the observers frame, Lorentz transformation will now distort those. Instead it is more natural to go back to Galilean transformations which preserve these. But looking at Maxwell in vacuum, this now means that this equations looks like a classical wave equation in a medium. Furthermore gravity itself will also appear back as a force field (and some other additional fields). An increase in complexity which is compensated by trivializing the geometry. Conceptually this is similar to a reversed Newton-Cartan approach, where we take general relativity and translate it back to a Newtonian-like "absolute time" (which TDB is).