Breaking the Universe's Speed Limit

Quantum gravity depends upon inertial and gravitational equivalency in keeping with space that is made equally larger and smaller in keeping therewith. This balances attraction and repulsion, as the space is BOTH flattened/contracted and stretched/expanded in a balanced fashion.

Look at the ground and your feet. Gravity is key to distance in/of space.

I already proved this in/as dream experience. The center of the body is the linked origination and generation of our experience.

You all are repeatedly going about the unification of physics in the wrong manner.

DREAMS COMBINE AND INCLUDE OPPOSITES.

Hi Edwin,

If we are going to talk about Atlas, have you heard the rumor on this diphoton bump with effective mass of 115 GeV?

Have Fun!

Ray,

Yes I have. What do you think it is?

Edwin Eugene Klingman

Hi Edwin,

It is where SUSY phenomenologists expect the Light SUSY Higgs. The strength of the diphoton signal is so strong that it doesn't look like a Standard Model Higgs. I suppose it could be a composite Techni-pion that provides a Goldstone origin of mass, but Technicolor has other problems that probably requires SUSY as a solution. The graph at the top of page 30 in this old paper with my colleagues and me shows how a relatively light-weight Light Higgs could exist, while m_1/2 (and most gaugino masses - most Neutralinos and Charginos) remain quite massive. This implies that we need to look closely for the Light Stop Squark.

The Tevatron has not confirmed this observation, although LEP had gotten to the edge of this region a decade ago, and thought they were on the edge of seeing something. The "leaking" of this paper did not follow protocol, and may or may not stand after the other couple of thousand LHC researchers "sign off".

I guess data and time will tell...

Have Fun!

The argument involves quantum uncertainty between different coordinates. Principally this is between time and position. Suppose there is a theory that time does not exist. All that exist with some geometric content is space. Suppose there is another theory where only time exists, but not space. It might then be that these two theories are complementary sets. They are complementary in the same way that position and momentum are complementary in standard quantum mechanics. If you read my article here on FQXI I indicate (though the calculations are not presented in detail as they are formidable) how light cones and Heisenberg groups emerge from a single structure. I will try to indicate how this works from more elementary considerations.

In 1930 there was a famous Solvay conference where Einstein and Bohr sparred over the reality of quantum mechanics. Einstein was convinced of reality and locality and argued staunchly for an incompleteness of quantum mechanics. Quantum theory could only be made complete if there are some hidden variables that underlay the probabilistic, nonlocal quirky aspects of quantum mechanics. At the 1930 Solvay conference Einstein proposed an interesting thought experiment. Einstein considered a device which consisted of a box with a door in one of its walls controlled by a clock. The box contains radiation, similar to a high-Q cavity in laser optics. The door opens for some brief period of time $t$, which is known to the experimenter. The loss of one photon with energy $E~=~\hbar\omega$ reduces the mass of the box-clock system by m = E/c^2, which is weighed. Einstein argued that knowledge of $t$ and the change in weight provides an arbitrarily accurate measurement of both energy and time which may violate the Heisenberg uncertainty principle ΔEΔt ~ ħ.

Bohr realized that the weight of the device is made by the displacement of a scale in spacetime. The clock's new position in the gravity field of the Earth, or any other mass, will change the clock rate by gravitational time dilation as measured from some distant point the experimenter is located. The temporal metric term for a spherical gravity field is 1 - 2GM/rc^2, where a displacement by some δr means the change in the metric term is ~ (GM/c^2r^2)δr. Hence the clock's time intervals T is measured to change by a factor

T --> T sqrt{(1 - 2GM/c^2)δr/r^2} ~ T(1 - GMδr/r^2c^2),

so the clock appears to tick slower. This changes the time span the clock keeps the door on the box open to release a photon. Assume that the uncertainty in the momentum is given by the Δp ~ ħΔr < TgΔm, where g = GM/r^2. Similarly the uncertainty in time is found as Δ T = (Tg/c^2)δr. From this ΔT > ħ/Δmc^2 is obtained and the Heisenberg uncertainty relation ΔTΔE > ħ. This demands a Fourier transformation between position and momentum, as well as time and energy.

Consider an example with the Schwarzschild metric terms. The metric change is then ~ 1x10^{-12}m^{-1}δr, which for δr = 10^{-3}m is around 10^{-15}. Thus for a open door time interval of 10^{-2}sec, the time uncertainty is around Δ t ~ 10^{-17}sec. The uncertainty in the energy is further ħΔω, where by Fourier reasoning Δω ~ 10^{17}. Hence the Heisenberg uncertainty is ΔEΔt ~ ħ.

This argument by Bohr is one of those things which I find myself re-reading. This argument by Bohr is in my opinion on of these spectacular brilliant events in physics.

This holds in some part to the quantum level with gravity, even if we do not fully understand quantum gravity. Consider the clock in Einstein's box as a black hole with mass m. The quantum periodicity of this black hole is given by some multiple of Planck masses. For a black hole of integer number n of Planck masses the time it takes a photon to travel across the event horizon is t ~ Gm/c^3 = nT_p, which are considered as the time intervals of the clock. The uncertainty in time the door to the box remains open is

ΔT ~ Tg/c(δr - GM/c^2),

as measured by a distant observer. Similarly the change in the energy is given by E_2/E_1 = sqrt{(1 - 2M/r_1)/(1 - 2M/r_2)}, which gives an energy uncertainty of

ΔE ~ (ħ/T_1)g/c^2(δr - GM/c^2)^{-1}.

Consequently the Heisenberg uncertainty principle still holds ΔEΔT ~ ħ. Thus general relativity beyond the Newtonian limit preserves the Heisenberg uncertainty principle. It is interesting to note in the Newtonian limit this leads to a spread of frequencies Δω ~ sqrt{c^5/Għ}, which is the Planck frequency.

The uncertainty in the ΔE ~ ħ/Δ t does have a funny situation, where if the energy is Δ E is larger than the Planck mass there is the occurrence of an event horizon. The horizon has a radius R ~ 2GΔE/c^4, which is the uncertainty in the radial position R = Δr associated with the energy fluctuation. Putting this together with the Planckian uncertainty in the Einstein box we then have

ΔrΔt ~ (2Għ)/c^4 = L^2_{Planck}/c.

So this argument can be pushed to understand the nature of noncommutative coordinates in quantum gravity.

So these arguments concerned with the existence of time are interesting in some sense. I have not been a particular partisan in either say, where Fotoni argues time exists, but not space. Of course space and time are really just holographic manifestations from strings and Dp-branes, which means space and time have no degrees of freedom of their own. However, it does seem that it is possible that space and time are complements in a quantum mechanical sense within quantum gravity. The speed of light is something which emerges from this parabolic group structure I work with which derives Heisenberg groups and light cones.

Cheers LC

LC,

Thank you very much for your excellent reply to my question. You have explained this as clearly as I could have asked or hoped, and your answer is very logical, indeed. Moreover, I'm certainly not in any position to pick an argument with anything you've spelled out here.

That said, however, I'd like to clarify what I see as being probably the primary reason why some of us who are part of what I've come to think of as "the FQXi Time Mafia" seem to be talking past one another when it comes to discussing these things. That reason is, in my opinion, that we are coming at the topic of time from two rather dramatically different paradigms regarding the fundamental nature of time. These different paradigms naturally lead to different ways of interpreting the same empirical observations.

I'm not trying to say that either of these paradigms is necessarily more "correct" than the other, only that they are different and lead to different ways of thinking about the same things, just as there are different ways of interpreting what we're seeing when we look at a Necker cube, for example.

One paradigm for the nature of time, the mainstream, prevailing paradigm, which is a key component in the foundation of modern physics, including especially special and general relativity, is most succinctly summarized, in my opinion, by what has been called the operational definition of time, i.e., time is that which is measured by clocks. Taking this view as a starting point, everything else which is taken as orthodox thinking about physics follows logically and consistently (certainly for the most part, at least).

The other paradigm is the one which I've attempted to spell out in purely qualitative terms in several essays such as those here and here. I don't even know what to call this paradigm. Perhaps it could be called a relational paradigm for time? In a nutshell, it holds that particular times correspond with, and are identically equivalent to, particular configurations of the universe. This view takes clocks to be specific subsets of an evolving universe; in this view, the special role given to clocks in the mainstream paradigm is viewed with skepticism. Far from being a magic bullet for resolving all of the outstanding conundrums of modern physics (see Lee Smolin's 'The Trouble With Physics,' for example), this paradigm raises a host of knotty problems and questions of its own, and I have answers to none of them.

I do not know how to describe this "relational paradigm" in a more rigorous fashion, but it is possible that another, far more clever, person, Joy Christian, has made a stab at this in an essay here. I would not presume to speak for Mr. Christian on this point; it is possible that he would totally and vehemently disagree with me.

Regardless, thank you for engaging seriously on this topic, LC.

Regards,

jcns

J.C.N. Smith,

The nature of time is an interesting subject. I don't get very partisan over the issue of time existing, for physics seems not to welcome ontological or existential ideas from the outset, but only suggests these within some theoretical construct. We have some issues with reality in a quantum mechanical content, in particular with nonlocality and a local definition of reality. Time and space though are not impacted by this. However, it is likely that quantum gravity will have implications along these lines. Julian Barbour is pretty much into the notion that time does not exist. This is based largely on the Wheeler DeWitt equation HΨ[g] = 0, which is a quantum version of the Hamiltonian constraint in ADM relativity.

I could well enough imagine presenting how time exists, but space does not. We could presume there is some one dimensional space, a line or curve, and there is a fibration on that space by a three dimensional space. This internal space is a symmetry of the dynamics of this one dimensional parameterized space we label as time. This then connects to relativity when we consider the metric line element

ds^2 = -c^2dt^2 g_{ij}dx^idx^j,

where mixed time-space metric components are not included. We have here two notions of time. The first is the proper time τ = s/c, which is the invariant of relativity. The other time is a coordinate time t, which is not an invariant.

The obvious question to ask is whether ds is real. We can multiply it by mc^2 and define an action according to the extremal principle of the proper interval

S = mc∫ds,

which appears real in some sense. It has units of action, or angular momentum, which is a measurable quantity. Yet there is something a bit troublesome about all of this. How does the observer on this world line actually measure this interval? A clock is employed which must have some system of oscillations, such as a spring. Yet this is measuring the invariant interval according to something carried on that world line that deviates from the world line. Hence some sort of nongeodesic motion is being used to define or measure an interval along a geodesic path. Of course I am thinking primarily of a mechanical clock, but an atomic one still appears to hold for an EM field must be applied to knock electrons in the Ce atoms.

This Lagrangian is measured according to something which is not invariant. So we might then consider that action as dS = pdq - Hdt. Now we have some Hamiltonian, which might include a part for the dynamics of the clock. Hamiltonians must be specified on some Cauchy surface of data with a coordinate time direction. Yet this has gotten us into some funny issue, for to define an invariant interval it appears that we need a coordinate defined clock.

So far we have some identification of Hdt, or the square of this, with the c^2dt^2 in the interval above. We then have that the bare action term ∫pdq is identified with

∫pdq = mc sqrt{g_{ij}dx^idx^j}.

So we have a bare action given by our fibration, but we also have some constraint, where H acts as a Lagrange multiplier. So we then have our one dimensional curve defined in a spacetime, where the space is the space of fibration and the Lagrange multiplier determines the symmetry of that fibration which is the Lorentz group.

Now to make things curious, we could imagine this picture as dual in some ways to the picture where time does not exist, but space does. The duality might then have a noncommutative coordinate geometric content in quantum gravity.

The acceleration can be found by a number of means. F = ma with the Newtonian law of gravity and the centripetal acceleration a = v^2/r,

mv^2/r = -GMm/r^2.

This gives v = sqrt{GM/r}, which is v = 29.5km/s or v = 2.95x10^4m/s, for the mass of the sun and r = 1.5x10^8km. The acceleration is then 5.8m/s^2.

As I indicated I think there is some noncommutative geometric issue with the nature of space and time in quantum gravity. This will be a noncommutative geometry that is more general than the Kahler geometry of geometric quantization. That extends the pseudo-complex symplectic structure of classical mechanics into a complex structure with a Hermitian complentarity between conjugate variables. A simplectic group with z_i = (q_i, p_i} (index notation implied) obeys dz_i/dt = Ω_{ij}z_j. In quantum mechanics this is generalized to a commutation system, and the symplectic 2-form implies an operator valued Hamiltonian. However, this is not the most general system possible. This may be extended to in noncommutative geometry with more general Usp(n) groups. A unitary Lie structure can give rise to commutators [q_i, q_j] = ħω_{ij}, which are necessary in string theory with uncertainty principles involving transverse and longitudinal string modes ΔX^ΔX^- ~ L_s = 4πsqrt{α'}, α' = string parameter and L_s the string length. Noncommutative geometry is then a setting for complementarity principles.

Since I drew this quote to make a point to Tom Ray, in the Clothes for the Standard Model Beggar post, I thought I'd repost it back here:

"The two cornerstones of modern physics, Einstein's general relativity, which explains the behavior of stars and planets on the largest scales, and quantum mechanics, which governs the interactions of subatomic particles, each paint a different picture of the role of space and time. General relativity weaves space and time together into a four-dimensional fabric that can be warped by matter, while the equations of quantum mechanics use an immutable absolute clock to measure out the regular ticks as time passes. This difference has led some physicists to ponder whether spacetime changes character on different scales."

My point is that while QM doesn't have an internal time and clock, because it treats everything as simultaneous, when making measurements, scientists measure one simultaneous configuration, then another. The consequence is that they inadvertently reintroduce Newton's absolute flow of time. On the other hand, as I keep saying, if we treat the quantum state as the constant, then it is the configuration which changes, thus it is a flow from future to past. This emergent time is mathematically relativistic, without having to propose blocktime. It does this by separating space from time, so that while space is a constant dimension, regulating the relationship between energy and mass, time is an effect of these relationships and entirely relative to them.

Amrit,

Congratulations on the article in Physorg:

http://www.physorg.com/news/2011-04-scientists-spacetime-dimension.html

Is c scalable? That's a worthy pursuit. I would like more details about the program for which you were awarded the grant.

A harder question might be what is not scalable. Is pi scalable? I would say no, but the ratio components are scalable. We think we know what is extensible and scalable by looking at networks. So, in this context, is a node scalable and thus emergent? A black hole is definitely scalable and we measure it in terms of star masses. Conserved quantities and invariant forms, constants of nature seem to go in lock step. Scaling an object seems to be the only way to normalize it. The size and shape of. say, a network matters at every node and stage of development. Are there certain nodes that are not scalable? Even when a node desires to limit its scalability could not halt the extensibility around it. The discreteness of the problem becomes elusive: at the very small, the kernel becomes degenerate, vanishing; at the opposite end, accountings must be made for stuff we can only observe indirectly (dark matter and energy). Must one conclude that energy is not scalable but a totality? If c is not a self dual of some ultimate form, is there some other normed space that can be constructed from dark matter and dark energy to give a sort of quantum Hall effect for c being scaleable? What kind of object will this "light" be? It will not be guage invariant if a rank two tensor, so it must be a tensor of higher rank. Where can I learn more details?

JCN

Yes, we are the "time mafia" and I am probably the worst of the bunch.`

I was tired of playing with shadows and ghosts. I devised a logical path for the reification of time directly from the requirement of a universe based on (or abiding by) the rule of non-contradiction. The passage of time is now a dynamic and substantial process that follows simple rules of logic which define both existence and causality. The passage of time makes everything and matter can be seen as a form of time replacing time passing by simple logical substitution.

Logic itself requires the ultimate reductionism, that, for the universe to be operational i.e. to work by itself, it has to be made of only one substance.

The "nature of time" is the most wrongly uttered words. The word "nature" calls for a specific aspect of time; what it is by itself, its ontology.

Looks simple and yet, I am the only one in the whole world who seem to be able to fathom this distinction. Everybody else is running around, wanting it all, but not willing let go anything...

Marcel,

Logic says we can't add apples and oranges. This means that logic only allows operations on elements of the same nature. So, an equation like E=mc2 can only be viewed as being logical if/if we realize and accept the fact that every variable and constant on both sides of the equation represent, at the most fundamental level, elements of the same nature and logically computable.

The reductionism in physics has followed our knowledge and has seen a reduction in the number of independent variables: electricity and magnetism, electro-magnetism, mass and energy, space and time etc. but, always gaining in the process the relation that tied them together. But physics limits the amount of reduction we can achieve because we must keep those dimensions around that allow us to do physics... We have to move right into metaphysics in order to effect the final reduction and gain, not one more relation, but the actual logical understanding of the universe. But for this, we have to let go of our reality, just for a moment.

The type of shotgun reductionism that Amrit proposes is about shooting down one important variable without gaining its proper relation to the rest of physics i.e. where it really fits into the big puzzle.

Marcel,

The Parmenidean view is similar in a way to the block universe view of time. The Heraclitean perspective is more in line with the ADM relativity view of time.

Cheers LC

LC,

Thank you for that clarification. I've tended to think of all versions of general relativity as being closely associated with the Parmenidean or block universe view of time, which is the antithesis of the paradigm for time which I've attempted to spell out in my various essays.

The disparity between the Heraclitean and Parmenidean views clearly is *not* an example of a distinction without a difference. The difference between these views (and what this difference means in terms which are meaningful to us as flesh and blood humans) could hardly be more dramatic, just as the difference between pre- and post-Copernican cosmology could hardly be more dramatic. And it is exactly for this reason that I'm puzzled (an immense understatement) that this question of which view is more "correct" has not been settled long before now. Using the more correct paradigm for the nature of time should, at least in principle, allow advances in science not unlike those which were made possible once the post-Copernican cosmology was accepted as being more correct than its predecessor.

I agree with David Deutsch that ". . . one of the most valuable, significant and also useful attributes of human thought generally is its ability to reveal and explain the fabric of reality." (D. Deutsch, 'The Fabric of Reality,' p. 3) Thus far, however, human thought apparently has not succeeded in finally resolving the Heraclitean-Parmenidean debate. This is astounding, in my opinion! And I place much of the blame for this failure directly on the faulty paradigm for the nature of time which has dominated western thinking since prior the advent of physics as a science.

The mainstream, prevailing paradigm for the nature of time has led to many brilliant successes, and this fact has tended to blind us to its less obvious shortcomings, in my opinion. I attempted to spell this out explicitly and as clearly as possible in my essay 'Time: Illusion and Reality.' Going back to the drawing board with a clean sheet of paper and reinventing physics using a different paradigm for the fundamental nature of time understandably is not something which would appeal to many of today's working physicists. And I certainly can't blame anyone for that. If I were more clever I'd give it a go myself, but I fear that the task far exceeds my abilities.

Regards,

jcns

Marcel,

Since Tom insists on questioning every term I use, I keep have to respond by further focusing my argument. I'll post this latest itineration and see if you think it makes sense, as a description of time:

There is only that thermal medium/particle cloud/quantum state.

As the energy/mass/plasma/waves/particles moves around, it changes configuration.

Since it is in a constant state of flux, these configurations are constantly being replaced.

Some areas in which there is greater activity naturally change shape faster than those with slower levels of activity.

Now since we are mobile points of reference, we are also one of those particles moving about. We are moving forward from our individual perspective and the larger situation is changing, so it is quite easy to conflate our sense of motion with this larger change, thus we seem to move from one configuration to the next, in much the same way we move along a path.

The larger reality though, is that our actions are balanced in this larger context, so there is that non-linear reaction to our motion, which compensates for our motion, to maintain the larger equilibrium.

Because there is just this sea of energy and energy is conserved, it is impossible to have sequential configurations co-existing, because they are constituted from the same energy.

So the old configuration fades into what we colloquially refer to as the "past." As all this energy bounces around, the new configurations emerge from this action. Those ranges of probabilities of what might happen as this energy interacts, are what we colloquially refer to as the "future."

So this current configuration emerges out of this action and is replaced by it.

What we colloquially refer to as the "present" doesn't move along some fourth dimension, because it is all that energy/mass particles/waves/thermal medium/cloud of motion that is all that exists.

Thus these moments of configuration emerge from that "future" and recede into that "past."

It is only because we can only exist in that present state, that we sense it as going from past configurations to future ones, but it is not the present which moves, only that it changes shape.

So while we have this subjective sense of "moving" from "past" to "future," the objective reality is that potential becomes actual and is replaced, ie. the future becomes past.

John,

I have great difficulty with the structure, format and content of the text.

What is well understood can be expressed clearly. I am not sure you understand well the idea you are trying to convey.

This appears to be the typical struggle when we mix elements of our reality and element of a real universe as can be deduced from various accepted theories.

1- future, present and past are true in our reality because we have no choice about it, this is the way we think and work. Period. There is no sense in screaming and kicking about it.

2- The real universe is different from our reality and can be inferred/deduced from literal understanding of SR & GR. and applying it to something that exists

in substance.

Again, contradiction and paradox appears when we mix and compare elements from our reality and elements from the real universe.

I am always interested in a theory of time that addresses a specific component of time. I always do it; try to re-formulate it in simple terms ..

The Perception of time I leave to psychologist and neurobiologists and maybe some philosophers etc.

Marcel,

To JCN Smith,

Einstein probably made the biggest push forwards in our understanding of time. He demonstrated with the invariance of light speed that time was interchangeable with space. This is the Lorentz transformation. This extends to general relativity, where now spacetime is in a sense the "field" of gravity. However, something is odd here, for gravitational degrees of freedom are contained in the Weyl tensor. So the field of gravity has physical degrees of freedom in the case of gravity waves where C_{abcd} =/= 0. In the case of a black hole, a stationary observer may watch clocks on objects falling into the black hole slow down to a stop as they asymptote to the horizon. However, this is not a system which interacts with the exterior world by gravitational degrees of freedom in this stationary configuration. An object fall towards a black hole is observed so the frequency of radiation from the body decreases as it red shifts "to infinity," which is the same as watching time slow near the black hole. What does change, or is not entirely static, is the fact that a body approaching a black hole adjusts the mass of a black hole. As a result the stationary (noninteracting) degrees of freedom associated with a black hole changes. This is a sort of gravitational form of the measurement problem.

Time may then be a one dimensional aspect of reality that reflects how gravitational degrees of freedom transform. In that setting time is something akin to a one dimensional communication channel, or a form of transmission line. It is then possible that time is a parameterization of a quantum error correction code for the universe, where the most general form might be the Jordan matrix algebra.

Jacob Bekenstein and Avraham May demonstrated how black holes are one dimensional channels of quantum information back in 2001. Ashoke Sen demonstrated something similar to this. An anti-de Sitter (AdS) spacetime of dimension n is equivalent to a conformal field theory of dimension n-1. The isometries of the AdS spacetime are equivalent to the conformal symmetries of a conformal field theory (CFT) on the boundary. The near horizon condition for a black hole in an AdS spacetime is AdS_2xS^{n-2}, where S^{n-2} is a sphere or dimension n - 2. The AdS_2 is a hyperbolic spacetime with a structure similar to the Escher prints of tessellated disks called circle limits. The boundary of this spacetime is CFT_1, in one dimension, where the isometries of this space are the group of conformal quantum mechanics. The group is the set of diffeomorphisms of the circle (which bounds the AdS_2 disk) and this defines the set of bosonic string states. The AdS_2 is then equivalent to a circle group on CFT_1 ~ S^1xS^0 (where this is a generalization of CFT_n ~ S^1xR^{n-1}) and the S^0 defines two points. These two points turn out to be dual states for the Hartle-Hawking vacuum (HHV). The HHV comes from their seminal paper in 1984 on the wave function of the universe.

The next interesting direction is that Borsten, Duff, Marrani, and Rubens published a paper which illustrates a mathematical correspondence between entanglements in 3 and 4 qubit quantum systems with BPS and extremal black holes respectively. To illustrate this it requires one work in the STU system with radial symmetry. In that case all the Killing isometries of the theory are timelike, and they are defined in one dimension of a moduli space. I have submitted for review a paper which connects this with the AdS theory above with AdS_2 ~ CFT_1, and a coset structure on the AdS constructs the counting of possible microstates on a black hole event horizon. This is a physical partition function, but it is equivalent to a partition function of the integers. This has been one of those unsolved problems in mathematics, which Brunier, Folsom, Kent, and Ono proved in January. This is the mathematics for counting up quantum states on a black hole, or equivalently on the boundary of an AdS spacetime. I have been working up a variant of the proof by Brunier, Folsom, Kent, and Ono, but where the partition function is derived by compactification condition with respect to the correspondence between multi-partite entanglements BPS/extremal black holes.

In the end time is a slippery topic, and how it is interpreted might reflect something of the appropriate field theoretic form of quantum gravity. As a result one has to keep an open mind on the topic, without making prior assumptions about the ontology of time.

Cheers LC

Lawrence,

"Einstein probably made the biggest push forwards in our understanding of time. He demonstrated with the invariance of light speed that time was interchangeable with space. This is the Lorentz transformation"

I think that a finite maximum speed of light gives us the time - space interchangeable relation, not its invariance. The speed of light may change from place to place ... But in any place, the speed of light is the local maximum speed.

For example, the light beam going to the moon and back changes speed a few times but in any point of this travel, it moves at the local maximum speed.

This is because the rate of passage of time varies locally between the Earth and the moon and c=m/s and those seconds don't run the same everywhere. This explains the local invariance; c is always adjusted to the local rate of time and the ratio is invariant. The measured value of c is invariant.

So, my point is that a local maximum speed is not the same thing as the invariance of that maximum speed relative to other various places. This said, it is locally measured as invariant, a constant, because we measure two aspects of the same thing; the local ratio of space-time to the rate of time.

This make sense to you?

Marcel,

Light cones one Earth, the moon and between the two are shifted a very tiny amount. However, given that time differences are ~ 2GM/rc^2 ~ 10^{-10} and that the light travels back and forth in about 2.5 seconds this is a very small effect. It would be on the order of a fundamental time unit of an atomic clock. Differences of this sort which involve the difference between ruler and clock distances are measurable for light or EM radiation passing by the sun have been measured.

LC