Hi Stuart, the coordinate time is the time as measured by the fastest running (though perfectly accurate) clocks of the universe. In a gravitational field, clocks slow down, and ditto for objects moving at relativistic speeds. So the coordinate time corresponds to that of stationary clocks distant from sources of gravity (but I think you already know that). Proper time is the time actually measured by a physical clock as it moves along some worldline. If you have happened upon a discussion of black holes, be extremely wary of what their proponents have to say about coordinate time: it does have physical significance due to the fact that no other clocks run faster. The laws of the universe cannot simply be ignored: objects native to this universe are forever bound by its physical laws.

Neither is invariant in general relativity, though each transforms in a different way under general coordinate transformations; whereas the proper time is invariant under global Lorentz transformations.

In any case, what matters is spacetime, not space or time separately.

What Dr Spivey is saying is that coordinate time is just proper time distant from sources of gravity, but I don't see how this specifies the coordinate time as opposed to the proper time at a point within the field of gravity. Dr Nicolis seems to be just saying that the coordinate time transforms as a coordinate, but again, this doesn't make it clear to me how to choose a the coordinate time in a specific way, as is done in the Schwarzchild solution which is used to explain gravitational redshift, u.a. I'd appreciate a little more clarification. Is there some specific mathematical condition ancillary to the basic equations of general relativity, or some unwritten rule, or is the attitude that the choice is arbitrary? (I'd disagree that the last attitude could be used to explain redshift in the Schwarzchild solution, as any choice of time coordinate other than the one used seems to--in my calculations--lead to a speed of light that varies with time, an unphysical result.)

Once more: coordinate time isn't invariant under global Lorentz transformations; proper time is.

Neither is invariant under general coordinate transformations.

In neither case does it matter how the coordinates, whether of time or space are chosen, since the invariant quantities don't depend on this choice.

The only thing that matters is to identify the invariants. If the spacetime is Schwarzschild, the only invariant quantities are defined at infinity.

Observer dependent quanities are, just, that. So it doesn't make sense to talk about the redshift, in general; it does make sense to talk about what particular observers measure. The observer at infinity, in Schwarzschild spacetime, for instance, is accelerating (at infinity itself the observer doesn't accelerate, but it doesn't detect the black hole spacetime, either, so the limit requires some care) lives forever-(s)he doesn't hit the singularity in finite proper time; the freely falling observer, in Schwarzschild spacetime, hits the singularity in finite proper time. And there do exist accelerating observers, in Schwarzschild spacetime, that, also, hit the singularity, in finite proper time.

There aren't any ``unwritten rules''. The rules are written out and described by the transformations that are symmetries. All observers, whose reference frames are related by general coordinate transformations, are equivalent; those that can't be related by such transformations, are not (in general relativity). That's why there's a physical difference between freely falling and accelerating observers: their coordinate systems can't be related by transformations that are symmetries. They define different invariants.

Hi Stuart. As you seem to appreciate, all clocks measure proper time, including those that agree with coordinate time. Therefore, there can be no distinction as far as their transformations are concerned. Nevertheless, coordinate time is special in that no clocks anywhere can run faster. Failure to comprehend the implications of this has in recent times given rise to the black hole information paradox, a fictitious problem pre-emptively solved by Einstein in 1939. He realised that there are worldlines in the hypothetical black hole spacetimes that cannot be extended arbitrarily without violating non-local constraints. That's why talk of hitting singularities in finite proper time can be safely ignored.

The statement that no clocks can run faster than ``coordinate time'' doesn't make sense, because it can't be given any meaning. There doesn't exist any globally defined coordinate time in either special or general relativity.

A freely falling observer, in a black hole spacetime, that has a space-like singularity, like Schwarschild's does, hits the singularity in finite proper time. So do some accelerating observers in such a spacetime. The value of the proper time depends on the observer and it's a standard exercise to calculate it. There do exist accelerating observers, however, that don't hit the singularity in any finite proper time. These latter observers define the black hole exterior, while the first two classes define the black hole interior.

These three classes of observers are distinct, because the reference frames they define can't be related by diffeomorphisms, the transformations that are symmetries of general relativity.

The so-called black hole information paradox, that pertains to how quantum matter could interact with curved spacetime, doesn't have anything to do with the topic, that's about classical gravity.

If that were so then claims that the universe is 13.7 billion years old would make "no sense" and cannot not be "given any meaning". What piffle. Actually there's nothing to prohibit the comparison of clocks to determine which has experienced the most elapsed time. The "standard exercise" of calculating how long it takes to arrive at a singularity is based on the extremely foolish assumption that the physical constraints imposed by the rest of the universe can be totally ignored. Wrong. In fact, calculations which are not the subject of dispute show that however long one waits, it is always possible to insert a mirror between an infalling photon and an event horizon so that the photon is reflected back out, proving conclusively that even pre-formed black holes cannot absorb particles in finite time. As Einstein knew, science is not about imitating others or mindlessly following dogma - it concerns the search for truth. By 1939 he had realised that all the black hole solutions to the field equations are unphysical. He can hardly be held resposible for today's madness where people work on problems solved long ago, people who consider themselves so superior that they will not deign to read or acknowledge his work, let alone try to understand it.

Stuart> ...the usual explanation of gravitational redshift (by the Sun, for example) seems to depend in an essential way on a particular choice of coordinate time (or am I mistaken about THAT?).

It depends on your definition of "usual". You may be correct that most explanations rely on a specific coordinate system, for simplicity. However, the effect can be calculated an any coordinate system, since it is a measurable physical effect:

Assume that the emitter sends out light with a frequency corresponding to transitions between two specific hyperfine levels in Cesium 133 atom (i.e., at a frequency ν of 9 192 631 770 Hz, https://physics.nist.gov/cuu/Units/second.html) at rest, and that an observer compares the received signal with the corresponding signal from a local Cesium 133 atom at rest, to find a redshift z such that 1+z = νlocal/νreceived.

This is quite simple to calculate in any coordinate system, making use of the fact that the Schwarzschild geometry admits a timelike Killing vector. It is less simple to explain what this is and implies. Another technical problem is to make sure that the position, and state of motion of the emitter and observer, are described consistently between various coordinate systems.

Robin> In a gravitational field, clocks slow down, and ditto for objects moving at relativistic speeds

Actually they don't, although this is not an uncommon way the describe the actual facts (which I regrettably also have done more than once, and which even more regrettably can also be found in some textbooks). All perfect clocks ticks at the same rate, as measured out by their eigentime!

A consequence of this is that the current definition of SI units, https://physics.nist.gov/cuu/Units/current.html, does not say anything about the altitude (i.e., gravitational potential) or velocity of the Cesium 133 atom used in the definitions of second and meter. I have not looked at the background information, but I guess they assume that (i) Cesium atoms for all practical purposes can be treated as perfect clocks, and (ii) the standard is to be applied in laboratories sharing position and state of motion with the Cesium atoms (so the latter should in principle be cooled down to a very low temperature).

And, since eigentime is the time ticked out by (almost) perfect physical clocks, it must also be invariant under coordinate transformations. Which it is found to be, by taking into account the transformation rules for the metric tensor.

Note added: What confusion statements like "moving clocks slow down" may lead to is well demonstrated by poor Herbert Dingle, http://www.abovetopsecret.com/forum/thread756819/pg1

"According to the special relativity theory, as expounded by Einstein in his original paper, two similar, regularly-running clocks, A and B, in uniform relative motion, must work at different rates.

....

and the question inevitably arises: How is the slower-working clock distinguished?"

To answer this question one would have to assume the existence of a preferred inertial frame, which actually was the pre-relativistic viewpoint.

What is true, is that moving clocks usually are described as ticking slower -- in coordinate systems where they they are described as moving. And they may, depending on the means and interpretations of observations, be observed like ticking slower --- when they are observed to be moving very fast.

When Einstein used to do thought experiment's for special relativity, he did it with two clocks. One clock kept time at the location where the journey started (coordinate time) and one clock traveled with the one on the journey (local time). Because of the effects of velocity on time, these clocks ran at different speed. The concept is kept for calculations in general relativity. For example, a particle falling freely to a black hole (if one could physically exist) is tracked using coordinate time--at a location at the start of the journey (e.g., so far away it is not affected by gravity) and local (now often called proper time) that is the time experienced by the particle as it continues on its journey.

Dear Dr Weller,

Why not just always use the time measured by the clock locally in whatever reference frame you happen to be in, just as you do in special relativity? (Not the time of the particle, unless you've transformed to its reference frame.) I.e.: why not just use t' = SQRT(g44)*t? This time would have no effect on the spatial metric or spatial distance, but would make the 4-metric time dependent & the speed of light would also be time-dependent, which is unphysical in the Schwarzchild solution, which is supposed to be a case of a static situation.

When one is studying the Schwarzschild metric, that's all there is-there's no ``rest of the Universe'', since the Schwarzschild solution is a vacuum solution of Einstein's equations and observers are probes in this metric. One can ask, whether perturbations of the Schwarzschild metric can affect it and this has been done. it's a standard exercise to include the energy-momentum tensor of matter, that isn't a probe, but can affect the metric and compute the result, in particular, whether the singularity can be affected. The answer is that ordinary matter can't remove the singularity. This doesn't mean that all observers can detect it-there do exist accelerating observers that can't detect any singularity at all. They do detect they're accelerating, however.

t' = (g44)^(1/2) t isn't invariant under general coordinate transformations, that's why it doesn't make sense as any sort of invariant time. In fact it doesn't even reduce to the proper time, when the metric can be taken as fixed. It's not possible to separate the spacetime metric in spatial and time-like part in an invariant way. All these issues are standard topics in a general relativity course, they're nothing new.

The statement that the Universe has a finite ``age'' is related to the question of spacetime singularities in the following way: The black hole singularity is a future singularity. Einstein's equations are invariant under time reversal, so it's possible to ask about past and future cosmological singularities and that's one way to define the Big Bang singularity. The way, what's called `the age of the Universe'', is actually determined, isn't that trivial and one presentation is here: http://w.astro.berkeley.edu/~dperley/univage/univage.html#cosmological

Indeed, the metric that does describe the Universe in accelerating expansion does have a past singularity and not a future singularity. A subtle point is that the measurements, that rely on electromagnetic effects, are limited, in the past, to the time when the Universe, as it expanded, did become ``transparent''.

Einstein's thought experiments can now be described much more simply, using parametric curves in spacetime. The relevant mathematical statement is that the solution of the geodesic equation, in a fixed (not necessarily flat) spacetime, by construction, maximizes the proper time; so any other curve between the same spacetime points, using the same metric, has a smaller proper time.

(Another standard subject in a course on general relativity.) There's no point mixing up history of physics with physics.

Stam Nicolis:

The moral is: first learn the physics; then, if you want to, go into its history.

The two are separate subjects, that don't have anything to do with each other, regarding the content. History of physics is personal, physics isn't. That Einstein, or anyone else, did something is of interest for the history of physics-not for physics, as such, that refers to what did Einstein actually do.

So, once more: proper time is the quantity that's maximized by solutions of the geodesic equation, in any fixed spacetime; coordinate time is just what the term means and nothing more. The solutions of the geodesic equation don't depend on how the coordinates of spacetime in general, and the coordinate time in particular, are chosen.

Dear Dr Nicolis,

I didn't say g44^1/2*t was invariant; I asked why don't you choose coordinate time to be the time reading on a clock in the reference frame of choice? This corresponds to choosing your coordinates such that g44 = c^2; i.e. choosing t' = g44^1/2*t where t is the time coordinate chosen by Schwarzchild. If you do this, the gravitational field of the Schwarzchild solution becomes nonstatic (the g's depend on time) and the speed of light--defined as spatial distance/coordinate time along a light trajectory--becomes time-varying, all of which is unphysical in a static gravitational field. The question is, why choose your coordinate time for the reference frame as Schwarzchild did & not in some other way? In a GIVEN reference frame time and space CAN be separated & this is not invariant but specific to THAT reference frame. Schwarzchild's choice of x,y,z as SPACE coordinates specifies the reference frame, leaving only the time coordinate to be specified.

In special relativity, the coordinate time is the proper time along the trajectory of a particle at rest in the reference frame of interest plus or minus a constant of integration dependent on the world-line (I call this constant the "temporal drift"; for constant velocity, it is generally set to -vx/(c^2(sqrt(1-v^2/c^2)). My question is, apart from the choice of the temporal drift, why isn't this procedure followed in general relativity?

   This is as clearly as I can state the problem:-

     In SRT coord. time = integral of interval ds (= "proper time") over a trajectory of a point at rest in the reference frame of interest (in the rest frame that would be sqrt(g44)*t + constant of integration): WHY NOT DO IT THIS WAY IN GRT?

     Though ds is not a complete differential, it CAN be integrated over a SPECIFIC TRAJECTORY (is path dependent, i.e. time is relative to the observer) & this IS done in SRT--WHY NOT IN GRT?

I think reading the history of physics is very important to understanding how the mathematics of physics is connected to the real observable world. For example, in 1908, Minkowski wrote a paper on Space and Time. In it, there is what he calls "the mystic formula", that purports to show a connection between distance and time. It is actually a point of disconnect between mathematics and observable reality. Einstein followed Minkowski in his 1916 seminal paper on general relativity, and Schwarzschild derived his metric based on Einstein's 1916 paper. The whole Schwarzschild spacetime inherits the mystic formula from Minkowski. The result is the part of the Schwarzschild metric that is "time like" is useful for calculations of physical phenomena. The part that is "space like" represents an area that cannot exist in the real world, and is wholly a mathematical construction. Without taking time to read the history, one would never realize the distinction.

Dear All,

I appreciate everyone's efforts in trying to help, but none of the answers really satisfied me, so I continued thinking about it intermittently, and about 30 minutes ago, it occurred to me that I had been tacitly assuming that in transforming from Schwarzchild's coordinates to ones using the same spatial coordinates but proper time instead of Schwarzchild's "t", that the proper time was independent of the space coordinates--i.e.: dt(proper)/dx = 0--and you can of course not do this without running into trouble with interchangibility of the order of differentiation. I think that if I continue looking into things from this point of view, it will finally resolve my difficulties--the nature of which I may not have specified clearly enough (even now), which obviously had an effect on your abilities to provide answers that would satisfy me. Sorry about this little transgression.

Thanx again to everyone,

Stuart Boehmer

   Actually, looking through my notes, I find I did not make the mistake I said I made in the last note, so I'm as perplexed as ever. I've decided to study the Schwarzchild solution in the limit M --> infinity, r --> infinity which turns out to be the case of a constant gravity field, which I find to be equivalent to the case of hyperbolic (constant acceleration) motion in SRT--appropriately enough (the principle of equivalence!). This simplifies the matter by removing the distracting complications of nonEuclidean space & space-time. I hope that by studying this simplified case meticulously I will be able to resolve my difficulties.

At last the fog has begun to part: I was concerned that if you transform from Schwarzchild's coordinate time to the physically meaningful proper time (as indicated by setting g44 = -c^2) that the other gmn become dependent on time giving the appearance that the gravity field is not static, which is absurd. The same thing happens for constant gravity as represented by constant acceleration, i.e., hyperbolic motion. So I was very perplexed. The idea is that an inertial frame infalling from rest at infinity (in the Schwarzchild solution) has a velocity dependent only on position--this is what's meant by saying it's a static gravitational field. Yet the gmn seemed to depend on t, i.e., a seemingly nonstatic situation. But it finally occurred to me that the same thing arises in Newtonian theory. For instance, in a constant gravity, you can write V = gt, which depends on time & appears to be nonstatic, yet a constant gravity field is certainly static. The key is to write V as a function of x, which, since, x = gt^2/2, means, V = SQRT(2gx). I'm not sure if I'm being clear, but at last I'm satisfied that if I pursue the analogy of these ideas in the relativistic case, it'll clear up my difficulties.

Dear All:-

   I apologize in advance if this missive seems confused and rambling, but I am still in an unsettled state of mind about the whole issue.

     Basically, I've accepted the party line.

     That is to say, Schwarzchild's time, t, is simply a coordinate--i.e. merely a timelike parameter, though not the physical, "proper" time, given by the 4-interval along any trajectory. (Trajectories can be parameterized by any timelike variable, it is not necessary to use the time, even in Newton.) Because of the general covariance of the theory, we can be assured using it and transforming the solution into terms of the real time in the last step gives the same results as using the true time directly--which would be computationally more complicated as the space-time metric is no longer diagonal, and depends on time as well as space coordinates. But it certainly is eyebrow-raising that a "static" gravitational field, when expressed in terms of the true time (i.e., the space-time metric, which presumably represents the strength of the gravitational field) is time-dependent. But maybe this isn't so serious, as I'm not completely certain what a "static" situation in nature is anyway. Gravity cannot be cancelled out by any other force (e.g., electromagnetism) because a force is a vector, while gravity--the equation of motion for free fall--involves a term involving the Christoffel symbols, which are not vectors, but depend on the second derivatives (i.e., basically, the acceleration) of the transformation, so a state of perfect rest in a gravitational field is not possible--thus the surface of the sun is constantly boiling, the atmosphere of the earth is slowly evaporating, etc., etc. So I guess we just have to be satisfied with the definition that the condition is "static" when the spatial metric is independent of time--which it is in the Schwarzchild solution, no matter what variable you use for time. Of course, the spatial metric being independent of time is the condition of rigidity--so the spatial geometry of the reference frame of a static gravitational field must be rigid, as Schwarzchild's is--something that also seems unlikely to occur in nature.

     In my interpretation, the space-time metric is merely a tool for calculating the proper time along any trajectory & could be eliminated if we computed the proper time from direct relations in coordinate transformations, if we insisted a physically meaningful theory could use only physically meaningful parameters, such as the proper time. This is perhaps a mathematically immature attitude--perhaps we should be opportunistic & use any parameters that simplify the problem, but it makes for poor philosophy (which concerns me also) in presenting the fundamental form of the theory, which, I believe, should be philosophically clear about its physical meaning, and not guided by mathematical pragmatism. This may be old-fashioned in view of the existence of a well-established theory called quantum mechanics, but still...

    Anyway, I have some vague ideas on how to eliminate the space-time metric from GRT, and just get by with the spatial metric and a fundamental velocity field, associated somehow with the (inertial) reference frames in free fall. I call this space with time--as opposed to the "space-time continuum"--i.e., making space dynamic, by introducing time into geometry, rather than the Minkowskian concept of making nature seem static (which it certainly is not) by ossifying time by reducing it to merely spatial/geometric concepts (the "block space-time" so popular among those who interpret SRT to mean that "time doesn't really exist").

    Maybe one day I'll get around to completing this ambitious project--the reformulation of SRT & GRT, and a generalized theory of a time-dependent geometry, (something akin to Riemann's accomplishment), which could be applied to Newton's theory as well as relativity, or more general geometries having nothing to do with physics--a purely mathematical animal.

After thinking about this problem since originally posting it about 1 year ago in Jan., 2018, I've finally arrived at certain conclusions which I've written up in a paper called "The Theory of the Relativity of Simultaneity" for the all-access journal Theoretical Physics, a publication of Isaac Scientific.

My intuition that the physically meaningful time coordinate leads to g44=-c2 was correct.

In the case of hyperbolic motion (i.e., constant acceleration, which is equivalent to constant gravity by the principle of equivalence) this leads to a non-time-orthogonal (gm4 not zero) 4-metric which is time-dependent; this appears problematic as a constant gravity field is a static physical situation & therefore should not be time dependent, and the received wisdom is that the 4-metric of the 4-interval is the "potential" of the gravitational field.

But now I believe that the received wisdom apropos the physical significance of the 4-metric (& the Minkowski space-time concept generally) needs to be re-thought.

In the paper I show that the spatial metric (the 3-metric) is Cartesian & independent of time, while the time-dependence of the gm4 components of the 4-metric represent gravitational time dilatation or red-shift. Thus, I conclude, the proper quantities representing the gravitational potential are the spatial metric and the time derivatives of the "temporal drift" (the gm4 components of the 4-metric when g44=-c2 ).

I now believe that the theoretical foundations of the general theory of relativity need to be completely re-formulated (without necessarily changing its content). Currently, it is based upon the Minkowski conception of a 4-D space-time manifold in which the fundamental quantities are the 4-metric and the 4-interval and in which the principle of relativity is represented by covariance of the laws of physics in terms of transformations between coordinate systems/reference frames. All this needs to be re-thought completely.

ANY IDEAS? I'd appreciate hearing from anyone with ideas on all this.

My current thinking is along the lines of a concept of "space with time" (as opposed to a space-time continuum) in which the spatial distance & the time interval along a trajectory are considered separately rather than through a single quantity (the 4-interval of space-time).

This idea, I believe, is more physically meaningful than the usual conception of a Minkowski space-time continuum as well as being more robust as the latter idea falls apart completely if the Lorentz factor for the time dilatation of a clock moving with speed v with respect to an inertial reference frame is not precisely SQRT(1-v2/c2).

See my paper if it is published by Isaac Scientific for further details.

For my latest musings on this topic--in which I stick stubbornly to the principle that the time coordinate should be equal to the reading on a clock, and not a meaningless coordinate--see my post "The Theory of the Relativity of Simultaneity" in my pre-prints (it is a work in progress).

For a definitive answer to this question, see my paper under "pre-prints" called "Radially Falling Reference Frames in Gravity."

The most common description of the electromagnetic field uses two three-dimensional vector fields called the electric field and the magnetic field. These vector fields each have a value defined at every point of space and time and are thus often regarded as functions of the space and time coordinates. As such, they are often written as E(x, y, z, t) (electric field) and B(x, y, z, t) (magnetic field). To bring such a manner of descriptive Mathesis into Relativistic spacetime (algebra) would most naturally necessitate transformations on the field continency of those elements that could potentially de-harmonize some of the most basic elements put into all four Maxwell’s equations for the sake of bringing out the medium-less field characteristics of electromagnetic waves . To accomplish this , we cannot simply add the four linearly . Function space. In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. In fact, Lorentz, Poincaré and others already knew that they needed to apply the Lorentz transformation to Maxwell’s theory of classical electromagnetism, and had succeeded a few years earlier in formulating a theory which is extremely similar to Einstein’s in its predictions. Some important experimental verification of this was also available before Einstein’s work (most famously, the Michelson-Morley experiment). The frame has actually an origin where the average motion of all the universe’s galaxies is zero. If a clock were fixed there, it would be motionless while the universe expands around it. This is a unique, privileged reference frame for astronomical convenience, but there is no reason to suppose it is otherwise privileged. It is not the frame sought by the A-theorist who believes in a unique present, nor by Isaac Newton who believed places can be at absolute rest, nor by James Clerk Maxwell who believed in an aether that was supposedly at rest everywhere in the universe except that it shakes as a light wave passes by. Time is represented as one-dimensional in all the fundamental theories of physics. What I am doing is considering it as zero-dimension of all numerical progressivity of whatever processually established phenomena that have the capability to apply real numbers for purposes of procreating a continuum, as is the time line in a typical reference frame used to represent time’s topology (because “is temporally less than” is a transitive, connected, and antisymmetric relation). As a result , consciousness requires a grouping of elements within a system of four-elemented transformativity that have physical cause-effect power upon one another.