There are three levels to consider.

If one agrees that 4 coordinates (whatever they are, if they are compatible) are necessary to locate an event, the universe (the container in which we live) has the structure of a manifold. A manifold can be represented in many charts. This manifold structure has a particularity : one of the dimension is singular : one cannot travel in time (so that we cannot measure time with the same procedure as lengths), and an observer can locate events belonging to his present. This lattter feture creates a folliation of the manifold in slices (the present of the observer) in 3 dimensional hypersurfaces (the usual space,at a specific time of the observer). But this folliation is specific to each observer.

The second level comes with the tangent space,to measure vectors, tensors, and derivatives. For this you need a frame, and obviously you can choose 3 spatial vectors, but the 4th vector must be imposed : it is in the direction of the motion of the observer in the 4 dimensional universe. Once more it is specific to each observer, but because nobody sees things moving towards his past, there is someting special about the velocity of the observers. This feature is explained in the 3d level.

The third level is the metric (it is not necessary for the first two). When measures are done in orthonormal frames, the frames define a scalar product, (that is a metric) and the changes of frames which preserve the scalar product are given by an orthogonal group. Orthogonal matrices have 4 real eigen vectors, and it is obvious that all 4 are not identical : one of them is special, and this gives the two lorentz groups with signature 1,3 or 3,1.

So, on 3 counts, relativity does not put time on the same footing as the others. On the contrary.

I don't think I can agree with your paper, but maybe I misunderstood parts of it so please do bear with me.

1- first, you state - if I understood you correctly - that the Schwarzschild metric is dependent on the coordinate system employed. It is absolutely not. It is the envelope hull at which the escape velocity becomes equal to that of the speed of light - simple as that. It has nothing to do with coordinates and all to do with gravity intensity. The equation for escape velocity calculation is very simple - it's the point where potential energy balances out kinetic energy.

2- Second - I'm aware that this this might be only a matter of terminology- you keep speaking of a flow of time, which is not a proper description, certainly not in a text that treats time as a dimension interchangeable with the other dimensions of spacetime. (This would not be the only terminology inaccuracy, you use the word 'fade' when it is clear that you meant 'fed' for instance.)

3- More importantly, the premise - if I understood your text properly - that time remains an invariant under all coordinate systems is incorrect, and seems to lead you (if I understood correctly) to circular reasoning: i.e., relativity is wrong because relativity is wrong. For instance, take a constant-speed sliding coordinate system. There is a counterintuitive interplay of speed, distance, and time, in relativity, so that if you move towards a target at constant speed, for instance, your simultaneity with events happening at that target is not the same with respect to what it would be if you stood still, or if you moved away from that target. This is exactly what Einstein's equations show (and this effect has been demonstrated experimentally), so that a delta T can come out differently between e.g. coordinate systems that move, even without acceleration, with respect to each other. Distance is also involved of course - it's a constituent element to the relativistic interplay in relativity, so that time somewhere cannot be apprehended the same way if your zero coordinate is on Andromeda or over here. Do the calculation, and you'll see that your simultaneity with events on Andromeda can shift by a whole day into tomorrow if you just walk towards it instead of standing still. Or into yesterday if you walk from it.

If you were right, then the consequences would lead to the crumbling of the edifice of relativity - but it has been experimentally demonstrated.

In more exotic terms - you can make a case that timelike and spacelike dimensions actually do swap when you enter a Schwarzschild horizon. And along with Itzhak Bars and others, you can certainly make a case that there are possible universes with signatures (N,X), with X higher than 1. Most of those universes would be unstable, except those were both numbers are higher than 3, in which case they're simply incalculable and unpredictable

Dear H Chris Ransford,

As you said " It is the envelope hull at which the escape velocity becomes equal to that of the speed of light - simple as that. It has nothing to do with coordinates and all to do with gravity intensity. The equation for escape velocity calculation is very simple - it's the point where potential energy balances out kinetic energy" 

But remember Einstein defined the escape velocity in GR in the Schwarzschild Geometry, as it is always classical escape velocity not relativistic. And if it is classical escape velocity, how and when it will equal to the speed of light? GR is not completely relativistic. Also according to the recent paper http://physicsworld.com/cws/article/news/2014/jul/28/new-correction-to-speed-of-light-could-explain-sn1987-neutrino-burst

treating light as a quantum object, the change in a photon's velocity depends not on the strength of the gravitational field, but on the gravitational potential itself. However, this leads to a violation of Einstein's equivalence principle – that gravity and acceleration are indistinguishable – because, in a gravitational field, the gravitational potential is created along with mass, whereas in a frame of reference accelerating in free fall, it is not. Therefore, one could distinguish gravity from acceleration by whether a photon slows down or not when it undergoes particle–antiparticle creation. What is you opinion????

Also relative to Pond-Rebka experiment; Proponents of the theory of General Relativity offer three different conflicting explanations of the results of the The Pound-Rebka experiment that are said to be equivalent to each other and therefore all equally correct. All make the claim that the results of the Pound-Rebka Experiment are “proof” of the Equivalence Principle even though nothing in these measurements suggests any need for the Equivalence Principle. What is you opinion??????

Dear Remi, I am sure you have not read the paper. I am well aware that spacetime is dynamic. The issue is, suppose I have my coordinates (x,y,z,t) and I -- for some reason best known to me; decide to instead to use (r,theta, phi, t) coordinates, is it correct to transform the time coordinate in the transition (x,y,z,t)--->(r,theta, phi, t); think deeply about this. Will a time duration of 5 minutes change to something else because of the transformation? If it does, this  means that the change of the label of space is a physical process. For all I know, changing of  the labels of space is not a physical process. Therefore, it is not only erroneous to transform the time coordinate but also erroneous to introduce a time dependence in the space coordinates as this is tantamount to a physical alteration of the system. This is the point that my paper is making.

Dear Remi, I never divorced time from space nor space from time. I am simply saying when it comes to coordinate transformations, time and space (space and time) are to be divorced completely for when joined together (as man has tried), clearly, they will preserve not an independent reality -- we will have what at best can be called magic. However, when it comes to transformations between reference systems (frames), time and space are joined together into an intimate union which no man can put asunder for any notion of space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

There are three levels to consider.

If one agrees that 4 coordinates (whatever they are, if they are compatible) are necessary to locate an event, the universe (the container in which we live) has the structure of a manifold. A manifold can be represented in many charts. This manifold structure has a particularity : one of the dimension is singular : one cannot travel in time (so that we cannot measure time with the same procedure as lengths), and an observer can locate events belonging to his present. This lattter feture creates a folliation of the manifold in slices (the present of the observer) in 3 dimensional hypersurfaces (the usual space,at a specific time of the observer). But this folliation is specific to each observer.

The second level comes with the tangent space,to measure vectors, tensors, and derivatives. For this you need a frame, and obviously you can choose 3 spatial vectors, but the 4th vector must be imposed : it is in the direction of the motion of the observer in the 4 dimensional universe. Once more it is specific to each observer, but because nobody sees things moving towards his past, there is someting special about the velocity of the observers. This feature is explained in the 3d level.

The third level is the metric (it is not necessary for the first two). When measures are done in orthonormal frames, the frames define a scalar product, (that is a metric) and the changes of frames which preserve the scalar product are given by an orthogonal group. Orthogonal matrices have 4 real eigen vectors, and it is obvious that all 4 are not identical : one of them is special, and this gives the two lorentz groups with signature 1,3 or 3,1.

So, on 3 counts, relativity does not put time on the same footing as the others. On the contrary.

Here is the idea; Since repulsion and attraction in electromagnetic mediated photons of spin 1, in this case the new equivalence principle must be related to changing velocity according to changing energy by absorption or loosing energy of frequency f. In this case Newton's second law must be defined in terms of frequency, that means it must be quantized and relativistic, and thus the inertial force must be defined as relativistic and quantized. In this case you get by the equivalence principle the escape velocity must be relativistic also and it depends on the amount value of the rest mass of the free fall particle under gravitational field that changed to photons which is equal to the potential energy, these photons let the free fall object to move in a relativistic speed equals to the relativistic escape velocity, and the relativistic mass always equal to the rest mass. In weak gravitational field this velocity can be approximated to the classical form of escape velocity, But in strong gravitational this speed is relativistic. GR is not completely relativistic. Review the Pond-Rebka experiment, and why the gravitational time dilation is equivalent to free fall object in gravitational field as moving with the escape velocity. Now what is the form of the Lorentz transformation to accept with this idea. In this case in each point in space Lorentz transformation must depend on a transformation from point to point in space, where it depends on the escape velocity at any point in space which equivalent to the potential energy at  any point in space where in this point the Lorentz transformation is linear . Because of that Franson calculated that, treating light as a quantum object, the change in a photon's velocity depends not on the strength of the gravitational field, but on the gravitational potential itself. http://physicsworld.com/cws/article/news/2014/jul/28/new-correction-to-speed-of-light-could-explain-sn1987-neutrino-burst

Now the form that the Lorentz transformation must take in this case; same form in x and t, But we must multiply y and z by the Lorentz factor in order to be agreed with the negative result of the Michelson-Morely experiment. In this case space is invariant and because of that universe is flat,  and the length contraction exists in the passed distance because of the effect of time dilation. in this case we keep on the Lorentz invariance but refusing the symmetry which leading to disappearing all the paradoxes in relativity, and we keep on the constancy of the speed of light locally. Globally there must exists a measured decrease in the speed of light in the gravitational field because of the effect of time dilation depending on the potential energy same as Franson''s calculated.

Dear Valentin,

I think that I understand your concern. What is time ? Too often people jump to complicated constructs, involving QM, photons,...without pausing one minute to look at the most basic and obvious questions , and puzzles, of our ordinary life. Mathematics are a very powerfull tool, but they will never answer a question such as : what is time ? Why are there 3 dimensons of space, and one of time (whatever the definition we give to the word dimenion). A formalism can just put on paper the song that we hear, so that perhaps we can replay it, but it will never tell us who is the piper and why he chooses to play that song and not another one. The goal of science is not to find the truth, or even to tell what is the real world, it is just to try to give the best, and most efficient representation of the world so that we can use what it offers. This is a noble goal, and it goes far beyond the usual tool kit of empiricists who, as seen so often, who plays with Hilbert spaces, eigen vectors and minimum variables substitution, ...This "modern" (100 years old!) Physics has lead scientists to avoid the most basic questions. Ask them what time is it ? and they will answer by saying that it is impossible to know because of the uncertainty principle.

This is sound to ask simple questions. And it is fortunate that we have now some good mathematical tools to express these ideas, because it enables us to meet the first criterium for a good scientifc theory : simplicity.. One of the image that I keep is of the train. There is often a train in narrative about relativity. We are seated in a car, and looking as the landscape goes by. It moves, it changes, this is a fact, I cannot go back, and this is the closest image that I have of the time. More or less what myriads of poets have told for millenia. This is because of this that I belive in the wold line, 4 dimensional universe. And Mathematics offer quite simple tools to give a drawing of the landscape. But why is it so ?

Dear Remi,

I am not sure to whom your thread is addressed...Anyway, I like trains, you like bats and caves, anyone can choose his own images. But to explain time from entanglement seems to me the a bit overstretched. You introduce metrology, and this is a good point. We measure elapsed time and distance between spatial points using two, very different methods and tools. How do we conciliate these measures ? We see that the ratio Length / Time has the dimension of a speed, and as all quantities in a formal system must be dimensionless (or at least using only a single unit), appears the need for a universal constant, which has the dimension of speed. That it happens to be c, the speed of light, is an experimental fact, this is not a necessity for the consistency of relativity. What is necessary is some fundamental assumption to compare the rate at which run clocks of different observers. And this leads, through the metric, because a metric is the base of any metrology, to the measure of both time and spatial distance. And then to the simplest (always look for simplicity) assumption that all material bodies and observers, on their train, go at the same speed. And you have almost all that you need to build GR.

OK, Remi, I agree with you.

You know if an area of space is flat, if an observer is at absolute rest, you know what entanglement means, what is a wave function and how it collapses, you knwo how to send superluminal signals. In front of such knowledge I can only surrender. I will keep my trains.

A coordinate system on a manifold has nothing to do with the metric properties of the manifold (or even whether the manifold has a metric). The metric of the 4-dimensional manifold of general relativity, "spacetime", is such that tangent vectors at each point can be classified as “spacelike”, “timelike” or “null”. That’s a property of the metric; it has nothing to do with coordinate systems. For convenience when dealing with particular solutions of the gravitational equations we usually find it  expedient to choose special coordinate systems so that at every point one of the coordinates corresponds to a timelike direction and the other three correspond to spacelike directions. The “principle of general covariance” does not have to respect that kind of choice. It means that the physics is the same in all coordinate systems. That's why it's called "general" covariance. It means that quantities and concepts are physically meaningful only if they are are coordinate-independent.

I agree with Mohamed El Naschie. If you were to change the minus sign in the Minkowski metric to a plus sign giving a metric ( +1, +1,+1, +1) this gives you what looks like Euclidean four Space as though time is like a special dimension. But this is strictly from a mathematical transformation of t -> It. In order to formulate Maxwell Equations that are invariant with respect to space-time coordinate transformations in this "new" space, it would now be necessary to change the Lorentz transformations somewhat. And it is also necessary to add a plus sign to the time-derivative term in the Maxwell equation expressing Faraday's law of induction. At first the results look exciting because you might just believe that there is a 'parallel universe' where particles can exceed the speed of light. But in the end, the solutions for electromagnetic waves are self damping in a vacuum and the Lenz law operates in reverse. The importance of the minus sign in the Minkowski metric is extremely important.