The fact that the speed of light is ( deemed to be ) constant is a clue for a quantity called a metric/ruler/interval, which by the manner of construction ( a quantity to calculate ) will also be constant. Using the notation from the previous post :

and so an interval of zero means we are talking of a photon path in spacetime, and this will be true regardless of which frame/observer used. The individual differences in space or time values will depend upon which photon ( transiting between events at the endpoints of a path ) and which frame you choose to measure that in, but the calculated interval* remains as zero for all photons.

Now for the relevance of this to inertial frame construction. Suppose I have already laid out a suitable network of markers for distance, in three dimensions. If you like visualise a 3D rectangular lattice**, like steel girders erected for a high-rise building but throughout all of space. There will be planes of constant x or y or z values. Where these intersect they do so at a right angle. While in practice that would be cumbersome/impossible to do so, at least we can imagine the fairly simple principles of laying out such a lattice. The question becomes how to set the clocks in such a scheme.

Remember the time of an event is the clock reading where ( in space ) it occurred. So let's put a clock at every desired lattice intersection, and we may ( linearly ) interpolate to gain any in-betweeners as we please. We may invoke anisotropy : there are no preferred directions in space. So for any two distinct spatial points the speed of light ( and the interval above ) is the same when traversed in either direction. So 'there & back' is equal to 'twice there' or 'twice back' for that matter.

I want to produce a neat animation for this ie. one common way to think of clock synchronisation. I had one some years ago but I lost it. I'll try to wrestle my copy of 3dsmax to the ground .....

Cheers, Mike.

* About which there is more than a bit of havering going on with regard to choices of terminology and definition. The end conclusions don't depend on said choices. I'm giving a common/simple variant ...

** It doesn't have to be of that type. But whichever it is must respect Euclidean geometry eg. using Pythagorus' Theorem. The nuance is that a reference frame may have several coordinate systems applying, with some problems finding one more useful/simpler than another. You could, say, have two systems of Cartesian type ( x/y/z ) differing in : point of origin; mutual rotation of axes; chirality ( left or right handed ). In any case a frame is the not the same concept as a coordinate system. It is legitimate to talk of frames without ever mentioning a particular coordinate scheme, indeed the tensor approach to GR has that as an advantage.

( edit ) You may be upset by my parenthetical comment for the speed of light "deemed to be". This is not as silly as it might sound. Firstly : any physical theory has assumptions which are subsequently validated by experimental test upon its deductions/consequences, and SR is validated in spadefuls. Secondly : this is what the Michelson-Morley experiment was all about, it established the simplest ( ie. see Occam ) explanation viz. no aether required and no speed variance found along all considered paths.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

Geometric Aside : higher dimensional Pythagorus' Theorem. Basically you want the square root of the sum of the squares, and herein lies a key geometric point.

You might ask : why did we not form

{ ignoring for the moment the value of c which I can set to 1 by changing units }

.... for the interval, that seeming to be a logical extension to four dimensions ie. just add a time dimension in with the 3 of length ? Well you could, but then we don't get what we want.

Let's step back for a moment. The general idea is that you can have a set of points. They can be 'indexed' or counted by n-tuples from some number set ( here the real numbers ). Each n-tuple is an ordered list of n numbers from that counting set eg. (-23, 45.7, 3, 157.2 ) and stands in place of each geometric point in some n-dimensional space. So there is a formal one-to-one correspondence b/w each point ( considered just as a geometric entity ) and a particular n-tuple. This is, of course, the concept of a system of coordinates that Rene Des Cartes broke the mould with and allowed algebra to assist geometry and vice versa. Now you can do a surprising amount of thinking and analysis using only this construction, without invoking any idea of 'distance' b/w points or their tuple counterparts. The core idea is to distinguish b/w the concepts of :

- geometric points

- their algebraic/numerical representation ( and for that matter their ordering properties )

- distance definitions

No doubt many of you have not made such a fine distinction because for normal/everyday life there is no reason to do otherwise. So it is all Euclidean/Pythagorean/Cartesian and leave it at that. However for four dimensional spacetime of Special Relativity that won't cut it. We use a non-intuitive distance measure ( metric ), and this has an opposite sign for the time variable compared to that of the spatial variables. That 'simple' aspect is what keeps SR 'sane', believe it or not, and neatly avoids the tricky 'when' questions that started this discussion off. Anyway this is Minkowski Spacetime and that is quite definitely not the same as some 4D Euclidean construct.

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

No luck with the animation, but kindly study this "God's Eye View" diagram :

which illustrates the progression of a photon there and back b/w two separated clocks that don't have any relative velocity ( eg. you could have a 'practically rigid body' spanning the space b/w them ). The approach is to set the right hand clock based upon the left one which we will call the master for discussion. We don't have to assume any congruence in their construction. We don't have to assume anything about how to get them to their positions, which is a keen issue as we may prove a change in the rate of time with motion using this construction! So this method to be described I call 'two-way in-place synchrony' which is Einstein's original logical construct. Also, strictly speaking, I don't have to assume that the clocks initially have the same rate of change ( see later ), but to speed up your revelations we'll assume they do. Now what happens, as a top down progression :

- clock A produces a photon, marked as *, at 0:00 ( an arbitrary but simple number to begin counting, if you're unhappy with that just trace through the logic with any other number that suits you ). Remember the time at clock A when that happened.

- the photon keeps travelling to the right.

- the photon gets to clock B. Two things will happen : (a) the time displayed on clock B is noted and (b) the photon is reflected back to the left.

- the photon keeps travelling to left.

- the photon arrives back at clock A and the time at clock A for that is also noted.

Now take the average of the two times at clock A. We want that number to be the one that should have been on clock B when the photon reflected. Here that is 0:05 ( but it could be some other number if you didn't want to start at 0:00 ). It wasn't of course, however we did write down that reflection time and so we now know how much to adjust clock B ( ie. the ? ). So here we have some signalling or whatever ( carrier pigeon if it has the stamina ) to transfer that adjustment information ( 'the photon reflection should have been at 0:05' ) across from A to B. Having subtracted ? from clock B I could re-run the same gag a few minutes later, say :

All of this assumes that we have 'regular' clocks. I will spare us the trip down that self referential rabbit hole. But suppose that when I re-ran at 4:00 onwards I found that the time on clock B when the photon arrived was not 4:05 ? You see clock B could again inquire of clock A what that photon reflection time should have been by the averaging at clock A, and any variance realised. For instance if the clock B reflection time was recorded as 4:05.1 I could easily deduce that clock B was running faster than clock A by 0.1 seconds per 5 seconds ( 0.02 seconds per second = 2% faster ). Presumably I could twiddle some aspect of clock B to match rates, plus re-setting the reading to once again come to synchrony. Rinse, repeat until you get to some suitable tolerance.

In this demonstration it is clear that the clocks are separated by five light-seconds ( ~ 15 x 10^{8 }metres ), because the round trip is ten seconds and anisotropy insists that "there & back" = 2 x "there" = 2 x "back". This works for any distance though, because the nett effect will be : if a photon travels y light-seconds then y seconds will be the time for that traverse. Recall my earlier insistence that event times are as per a clock at said event. Please also note that we are only setting up a clock network for use as part of a single reference frame/observer.

In effect we might claim that light travels at a constant light speed because our clocks & rulers are deliberately set up to generate exactly that result from any measurement ! :-)

{ And so it remains a matter of experience - that's what experiments do, generate experience - to see if that is a useful way to go about it all. }

Cheers, Mike.

( edit ) An aside you may ignore : imagine a world where we only used clocks and light speed to specify lengths. So all distances are quoted as so & so many light-seconds ( or other suitable multiple/fraction ). Suppose my house block is about 75 feet long by 75 feet wide using Imperial measure. Then at a rate of approximately one foot per nanosecond, I could quote that as being 75 light-nanoseconds long by 75 light-nanoseconds wide, or even having an area of 75^{2 }= 5625 square light-nanoseconds !

Hey Mike ! Nice block you have at 5.6K slns ......

or

Wow ! You have one whole cubic light-nanosecond of ice cream. Way to go ...

or as 12^{3 }= 1728 then that 351 cubic inch V8 engine is ~ 0.2 of a cubic light-nanosecond ( clns ). Etc .... LOL

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

So that is the gist of constructing an inertial frame as per SR. Now in the presence of gravity ( or frame acceleration ) the issue becomes keeping the clocks within a frame sychronised. That is, one can go through the process as described of two-way light signalling and get clocks to agree. But if you check again later there will be variance, more so the longer you wait before checking, and that will be so no matter how many times you repeat the process. If you like one could define non-inertial frames as those which display that clock behaviour.

Now you may know of the mutual clock slowing of inertial frames in relative motion. That is a between frame comment. I'm referring here to the 'simple' act of setting up a measurement system. You might object to the complexity of the discussion so far, in that we have examined in painful detail what you might deem as merely self evident. But Einstein made a point of challenging 'common sense' and therein lies his marvellous insights which have proven true upon testing.

It is OK to describe General Relativity as having constructable frames of inertial type, but you need a plethora of them to make progress. There isn't a single frame that encompasses some span of time and space for the purpose at hand. This was Einstein's idea of his Equivalence Principle. This translates to allowing an inertial frame to approximate some portion of a situation, but only over a limited spatial extent and only for a brief time interval. Exactly how much is 'some', 'limited' and 'brief' becomes, in the mathematics, a process generically called limiting arguments.

Try this for GR : call each little part of spacetime a 'box'. These are four dimensional. Each box we will model as being inertial ie. so that we may use SR within them in all it's glory eg. describing paths of objects etc. Adjacent boxes will differ in that they maybe considered as inertial frames with a relative motion. So we may look from one box to it's neighbour and use the SR transforms that demonstrate those oft quoted length contractions and time dilations. That means we have a way of describing the same phenomena from two slightly different points of view.

Perhaps you might imagine a light ray for instance that has a 'world line' or path in spacetime, or a material body winding it's way through those four dimensions. This is a global view for which a single inertial frame will not suffice. These objects travel along their world lines. Or if you like the world line is the object. Each such curve is composed of tiny boxes strung together like beads on a string. For a given actual curve segment we might divide it up into a certain number of boxes and we can consider pairs of adjacent boxes. If we increase the number of divisions ( more but smaller beads ) we make more boxes and pairs thereof. If we increase the number of boxes in such a way as to make them all become closer and closer together, then the smaller is the deemed relative velocity between each pair and the more gradual the SR transform undergone between each member of a pair.

Now we want to divide the path into an infinite number of infinitesimal boxes. Each of these boxes will get the fancy title of Momentarily Co-Moving Reference Frame ( MCRF ), and a path will be the concatentation of such. Here is where ordinary language will likely/quickly fail us and one has to transition to the more precise formalism of mathematics. To be exact the topic of differential calculus with an overlay of tensor notation.

The differential calculus deals with those 'rates of change' or transforms between the wee boxes. Determining a rate of change is called differentiation, and is much like deciding what gear you have your car running in at some moment. There is a ratio b/w engine speed and road speed that a specific choice of gearing decides. Thus when one applies the SR transform for length contraction ( Lorentz ), that is a rate of change ie. what ( mutual frame velocity dependent ) number does one apply to correctly describe such length alteration.

{ There is a cheeky aside to this. We can say the object is unchanged but the ruler varies. This is the essence of the metric concept and underlies much of the commentary about bending/curvature in space and/or time. }

The tensor notation is one way of managing the complexity of the mathematical expressions. The Lorentz formulae in their most general form have each of the x, y, z and t coordinates in one frame depend upon all of the x, y, z and t coordinates of another frame. One could organise this complexity in, say, a matrix form. But that has some unhelpful limitations I won't go into. The tensor forms have a snappy brevity which requires not inconsiderable effort to understand ( I am by no means fully conversant in this ), but does correctly condense all that matters in GR.

Suffice to say that there is an exact methodology to precisely track all those little changes of metric as we move from little spacetime box to infinitesimally adjacent spacetime box. GR is one of those areas of knowledge where the principles are quite easy to state, not too onerous to speak of in generality, but pretty appalling ( for the uninitiated ) to handle in concrete instances.

Cheers, Mike.

( edit ) You can rewind the thinking the other way : GR should revert to SR as gravity diminishes. The wee relative frame velocities go to zero, the MCRFs become equal to each other, and you return to a single inertial frame ....

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

Now the deeper gag, the real deal as it were, is that nature behaves the way that The Relativities indicate even if we never set up all those clocks and rods. No known exceptions so far, despite ample testing. The 'failures' if any are due to limited misunderstanding/misapplication of the theory ie. using cake mix when you have a recipe for rabbit stew. You won't even get Rabbit Cake ! :-)

The clocks and rods construct is a way through the issues of 'what do you mean by length ?', 'what do you mean by time ?' and other geometric stuff eg. 'what do you mean by a right angle ?' Thus as you view some object descending toward a black hole's event horizon it will :

- never seem to quite get there,

- the light from it gets dimmer and redder in frequency,

- any regular motions it had seem to go slow-mo.

All of which could be summarised by saying that a clock travelling with the object appears to run slow by distant viewers. Or simply throw away the clock and just say time goes slower. Plus we also get around the awkward anthropocentric stuff ( cue solipsism ) that insists upon human presence for things to be real. Remember the use of the word 'observer' to represent the entirety of a coordinated data collection system.

{ Did the Universe not exist before the animal lineage now labelled as 'human' became self aware ? The whole cosmic background radiation field just happens to replicate the effect of a prior timeline before we became 'clever' ? Thus far at least SR/GR has a 'locality rule', meaning that I have to wait for something to propagate for cause and effect to occur. It may be that a supernova has occurred near us in the Universe but the light hasn't reached us yet. We could all be dead by tonite due to the radiation to be received one hour after I post this message. Who can say ? }

Finally : back to the observer on the planet Earth, with his gravitational force-vector meter, while looking at the Sun in the sky. Does the meter always point to the Sun ? Yes it does ( ignoring atmospheric refraction, a local irrelevance, or just go to the Moon instead ). Can you predict the course of the Earth from any chosen moment onwards by using the meter or the visual reference ? Yes, use either and the answer is the same. Can you likewise post-dict the current path/position of the Earth by using either ? Yes. Will there be a different apparent direction of the Sun in the future, with corresponding force components, due to evolution of the solar system as per known dynamics ? Yes. Etc.

GR achieves a more than adequate explanation of this by being an extrapolation of SR, which in turn factors in the light time delay from the get-go. That accounting occurs at the low level of data definition ( ... the time of an event is of the clock at the event etc ....). GR tags and transforms all the little adjustments required to track how measurements change across some system. The metric used is the one discovered that allows such transforms to be useful. It is a generalisation of the Minkowski metric.

Cheers, Mike.

( edit ) "Rene Des Cartes" : LOL! His middle name was Des ? :-)

( edit ) Any questions at all ? NB with the new web site there is no 'read' counter for posts and thus I could be talking to myself ... ah well, I could go all Matrixey about that. :-)

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

A common way to analyse the geometry of spacetime is to look further at the 'interval' :

Which, by the way it is defined here, will be a positive number if the distance components are greater than the time component, a negative number if the time component is greater than the distance components, and of course zero if the components cancel out. These are classified as space-like, time-like and light-like ( or null ) intervals respectively.

Thus a space-like interval b/w two events has more distance than can be crossed by a photon in the time available.

A time-like interval has more than enough time for a photon to travel b/w the two points.

Light-like/null interval has exactly the right separation in time and space for a photon to travel between.

In this language then, we were using null intervals to synchronise the clocks within an inertial frame, or if you like we made them that way by suitable clocks settings.

{ For the diagrams that follow I will suppress two of the distance axes of spacetime. Thus time will increase going upwards and the remaining spatial dimension is left-right. }

Now if you think of Newtonian spacetime ( light speed being effectively infinite ) then for a given observer at some moment, all of spacetime is divided into the 'past', the 'now' and the 'future' :

.... with 'past' and 'future' implying causation. I can affect the future ( anywhere in the cosmos after 'now' ) and the past ( anywhere in the cosmos before 'now' ) may affect me. But for Special Relativity we have :

Remembering that as time is that which is measured by a clock at an event, then 'now' is merely 'here & now'. The diagonal lines are light-like ie. photon world-lines intersecting at the 'here & now'. Future and past have the same meanings as in Newtonian spacetime, but there is a new category called 'elsewhere'.

Elsewhere is all of those events in spacetime that are neither in your past nor in your future. So over at Proxima Centauri things happen that we don't know about. Yet. In time we will be aware of them. Likewise we have things happening here & now, any effect of which will take time to get to Proxima. The duration of the elsewhere can be quite long, and more so the farther away in distance they are from our here & now :

If Event_A and Event_B are at Proxima then the duration b/w is nearly 9 years, if they are at the Andromeda Galaxy then the duration of elsewhere is about 4 million years. For our Sun there is some 16 minutes of elsewhere for each of our 'nows' here on Earth. If it's across my office then it's a handful of nanoseconds.

A famous example of elsewheres ( plural ) is the cosmic microwave background. Today this is reaching us from an entire universe's lifetime - approx 13.6 billion years - and corresponding distance away. We see photons as they were formed about 300,000 years after the Big Bang, but they've lost alot of energy ( about a thousand-fold ) in the expansion of the universe.

If you were Master Of The Universe and progressively increased the speed of light then SR spacetime would gradually become Newtonian :

.... in the limit of infinite light speed the past & future light lines become one, the 'now' ( across all of space ). An increase in the speed of light allows one to be affected by more events ( that were elsewheres and have become your past ) or to affect more events ( that were elsewheres and have become your future ).

{ Diagrams which only suppress one of the distance axes will give a conical shape that demonstrates the totality of light-like world lines. Hence the phrases 'past light cone' and 'future light cone'. Alas I continue to fail to draw 4D pictures ..... }

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

A famous example of elsewheres ( plural ) is the cosmic microwave background. Today this is reaching us from an entire universe's lifetime - approx 13.6 billion years - and corresponding distance away. We see photons as they were formed about 300,000 years after the Big Bang, but they've lost a lot of energy ( about a thousand-fold ) in the expansion of the universe.

A famous example of elsewheres ( plural ) is the cosmic microwave background. Today this is reaching us from an entire universe's lifetime - approx 13.6 billion years - and corresponding distance away. We see photons as they were formed about 300,000 years after the Big Bang, but they've lost a lot of energy ( about a thousand-fold ) in the expansion of the universe.

Where did the energy go?

SHORT ANSWER : Into separation of everything from everything else ie. gravitational potential energy which would be regained as kinetic if everything fell back together again. So the photons have lost energy emerging out of the common gravity well. In a sense the expansion of the universe is the ultimate 'stone throw'. How 'high' will it go ?

LONG ANSWER :While the above is true as far as we can determine for what we have observed, what adds to the confusion is where one places the baseline for energy accounting. Which in turn then relies on when/where a given theory attempts to fill in the blank space : <?!*WHAT'S IT ALL ABOUT THEN, WHERE DID THE UNIVERSE COME FROM*!?>. In the general literature and many pop science programs that substrate for discussion is frequently not plainly stated. By that I mean what/where is the entire definition of the thing we are describing ? Is our 'universe' part of another ? Is it all there is ? Are there multiple universes, some connected to ours and some not ? Etc.

One of the cuter theories is where the Universe is created by the collision of two higher dimensional objects called 'branes'. The energy to kick off the expansion is the energy in the branes. Such collisions, to those who are in the region of intersection ie. us, will look exactly like a Big Bang. So we are the products of an accident between higher dimensional vehicles and our energy is a converted form of what existed, 'kinetic' if you like, in those branes wandering about in some hyper-spatial whatever. That sounds all well and good but it isn't. You see I could follow Douglas Adams and say that the Universe was sneezed out of the nose of The Great Green Arkleseizure ( TGGA ). So we are the snot from TGGA's sinuses, and perchance if you were in that mucus it would all look like the Big Bang too. However neither has a specific observable signature that distinguishes them. So in a logical/semantic sense you can interchange one for the other and still see a cosmic microwave background etc. By their very construction the Braneworld and TGGA explanations are bound to agree with what we already know to be observably true. Neither are a physical theory of interest because neither proposes an observable phenomenon to test it's truth by.

The deeper problem here is that GR only dictates how universes change. It doesn't prescribe any specific states at any point. It just says : given this state A, then it will evolve to this other state B in such & such a manner.

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## Now see if I can get back on

)

Now see if I can get back on track .....

The fact that the speed of light is ( deemed to be ) constant is a clue for a quantity called a metric/ruler/interval, which by the manner of construction ( a quantity to calculate ) will also be constant. Using the notation from the previous post :

and so an interval of zero means we are talking of a photon path in spacetime, and this will be true regardless of which frame/observer used. The individual differences in space or time values will depend upon which photon ( transiting between events at the endpoints of a path ) and which frame you choose to measure that in, but the calculated interval* remains as zero for all photons.

Now for the relevance of this to inertial frame construction. Suppose I have already laid out a suitable network of markers for distance, in three dimensions. If you like visualise a 3D rectangular lattice**, like steel girders erected for a high-rise building but throughout all of space. There will be planes of constant x or y or z values. Where these intersect they do so at a right angle. While in practice that would be cumbersome/impossible to do so, at least we can imagine the fairly simple principles of laying out such a lattice. The question becomes how to set the clocks in such a scheme.

Remember the time of an event is the clock reading where ( in space ) it occurred. So let's put a clock at every desired lattice intersection, and we may ( linearly ) interpolate to gain any in-betweeners as we please. We may invoke

: there are no preferred directions in space. So for any two distinct spatial points the speed of light ( and the interval above ) is the same when traversed in either direction. So 'there & back' is equal to 'twice there' or 'twice back' for that matter.anisotropyI want to produce a neat animation for this ie. one common way to think of clock synchronisation. I had one some years ago but I lost it. I'll try to wrestle my copy of 3dsmax to the ground .....

Cheers, Mike.

* About which there is more than a bit of havering going on with regard to choices of terminology and definition. The end conclusions don't depend on said choices. I'm giving a common/simple variant ...

** It doesn't have to be of that type. But whichever it is must respect Euclidean geometry eg. using Pythagorus' Theorem. The nuance is that a reference frame may have several coordinate systems applying, with some problems finding one more useful/simpler than another. You could, say, have two systems of Cartesian type ( x/y/z ) differing in : point of origin; mutual rotation of axes; chirality ( left or right handed ). In any case a frame is the not the same concept as a coordinate system. It is legitimate to talk of frames without ever mentioning a particular coordinate scheme, indeed the tensor approach to GR has that

.as an advantage( edit ) You may be upset by my parenthetical comment for the speed of light "deemed to be". This is not as silly as it might sound. Firstly : any physical theory has assumptions which are subsequently validated by experimental test upon its deductions/consequences, and SR is validated in spadefuls. Secondly : this is what the Michelson-Morley experiment was all about, it established the simplest ( ie. see Occam ) explanation viz. no aether required and no speed variance found along all considered paths.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## Geometric Aside : higher

)

: higher dimensional Pythagorus' Theorem. Basically you wantGeometric Aside, and herein lies a key geometric point.the square root of the sum of the squaresYou might ask : why did we not form

{ ignoring for the moment the value of c which I can set to 1 by changing units }

.... for the interval, that seeming to be a logical extension to four dimensions ie. just add a time dimension in with the 3 of length ? Well you could, but then we don't get what we want.

Let's step back for a moment. The general idea is that you can have a set of points. They can be 'indexed' or counted by n-tuples from some number set ( here the real numbers ). Each n-tuple is an ordered list of n numbers from that counting set eg. (-23, 45.7, 3, 157.2 ) and stands in place of each geometric point in some n-dimensional space. So there is a formal one-to-one correspondence b/w each point ( considered just as a geometric entity ) and a particular n-tuple. This is, of course, the concept of a system of coordinates that Rene Des Cartes broke the mould with and allowed algebra to assist geometry and vice versa. Now you can do a surprising amount of thinking and analysis using only this construction, without invoking any idea of 'distance' b/w points or their tuple counterparts. The core idea is to distinguish b/w the concepts of :

- geometric points

- their algebraic/numerical representation ( and for that matter their ordering properties )

- distance definitions

No doubt many of you have not made such a fine distinction because for normal/everyday life there is no reason to do otherwise. So it is all Euclidean/Pythagorean/Cartesian and leave it at that. However for four dimensional spacetime of Special Relativity that won't cut it. We use a non-intuitive distance measure ( metric ), and this has an opposite sign for the time variable compared to that of the spatial variables. That 'simple' aspect is what keeps SR 'sane', believe it or not, and neatly avoids the tricky 'when' questions that started this discussion off. Anyway this is Minkowski Spacetime and that is quite definitely not the same as some 4D Euclidean construct.

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## No luck with the animation,

)

No luck with the animation, but kindly study this "God's Eye View" diagram :

which illustrates the progression of a photon there and back b/w two separated clocks that don't have any relative velocity ( eg. you could have a 'practically rigid body' spanning the space b/w them ). The approach is to set the right hand clock based upon the left one which we will call the master for discussion. We don't have to assume any congruence in their construction. We don't have to assume anything about how to get them to their positions, which is a keen issue as we may prove a change in the rate of time with motion using this construction! So this method to be described I call 'two-way in-place synchrony' which is Einstein's original logical construct. Also, strictly speaking, I don't have to assume that the clocks initially have the same rate of change ( see later ), but to speed up your revelations we'll assume they do. Now what happens, as a top down progression :

- clock A produces a photon, marked as *, at 0:00 ( an arbitrary but simple number to begin counting, if you're unhappy with that just trace through the logic with any other number that suits you ). Remember the time at clock A when that happened.

- the photon keeps travelling to the right.

- the photon gets to clock B. Two things will happen : (a) the time displayed on clock B is noted and (b) the photon is reflected back to the left.

- the photon keeps travelling to left.

- the photon arrives back at clock A and the time at clock A for that is also noted.

Now take the average of the two times at clock A. We want that number to be the one that should have been on clock B when the photon reflected. Here that is 0:05 ( but it could be some other number if you didn't want to start at 0:00 ). It wasn't of course, however we did write down that reflection time and so we now know how much to adjust clock B ( ie. the ? ). So here we have some signalling or whatever ( carrier pigeon if it has the stamina ) to transfer that adjustment information ( 'the photon reflection should have been at 0:05' ) across from A to B. Having subtracted ? from clock B I could re-run the same gag a few minutes later, say :

All of this assumes that we have 'regular' clocks. I will spare us the trip down that self referential rabbit hole. But suppose that when I re-ran at 4:00 onwards I found that the time on clock B when the photon arrived was not 4:05 ? You see clock B could again inquire of clock A what that photon reflection time should have been by the averaging at clock A, and any variance realised. For instance if the clock B reflection time was recorded as 4:05.1 I could easily deduce that clock B was running faster than clock A by 0.1 seconds per 5 seconds ( 0.02 seconds per second = 2% faster ). Presumably I could twiddle some aspect of clock B to match rates, plus re-setting the reading to once again come to synchrony. Rinse, repeat until you get to some suitable tolerance.

In this demonstration it is clear that the clocks are separated by five light-seconds ( ~ 15 x 10

^{8 }metres ), because the round trip is ten seconds and anisotropy insists that "there & back" = 2 x "there" = 2 x "back". This works for any distance though, because the nett effect will be : if a photon travels y light-seconds then y seconds will be the time for that traverse. Recall my earlier insistence that event times are as per a clock at said event. Please also note that we are only setting up a clock network for use as part of a single reference frame/observer.In effect we might claim that light travels at a constant light speed because our clocks & rulers are deliberately set up to generate exactly that result from any measurement ! :-)

{ And so it remains a matter of experience - that's what experiments do, generate experience - to see if that is a useful way to go about it all. }

Cheers, Mike.

( edit ) An aside you may ignore : imagine a world where we only used clocks and light speed to specify lengths. So all distances are quoted as so & so many light-seconds ( or other suitable multiple/fraction ). Suppose my house block is about 75 feet long by 75 feet wide using Imperial measure. Then at a rate of approximately one foot per nanosecond, I could quote that as being 75 light-nanoseconds long by 75 light-nanoseconds wide, or even having an area of 75

^{2 }= 5625 square light-nanoseconds !Hey Mike ! Nice block you have at 5.6K slns ......

or

Wow ! You have one whole cubic light-nanosecond of ice cream. Way to go ...

or as 12

^{3 }= 1728 then that 351 cubic inch V8 engine is ~ 0.2 of a cubic light-nanosecond ( clns ). Etc .... LOLI have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## So that is the gist of

)

So that is the gist of constructing an inertial frame as per SR. Now in the presence of gravity ( or frame acceleration ) the issue becomes keeping the clocks within a frame sychronised. That is, one can go through the process as described of two-way light signalling and get clocks to agree. But if you check again later there will be variance, more so the longer you wait before checking, and that will be so no matter how many times you repeat the process. If you like one could define non-inertial frames as those which display that clock behaviour.

Now you may know of the mutual clock slowing of inertial frames in relative motion. That is a between frame comment. I'm referring here to the 'simple' act of setting up a measurement system. You might object to the complexity of the discussion so far, in that we have examined in painful detail what you might deem as merely self evident. But Einstein made a point of challenging 'common sense' and therein lies his marvellous insights which have proven true upon testing.

It is OK to describe General Relativity as having constructable frames of inertial type, but you need a plethora of them to make progress. There isn't a single frame that encompasses some span of time and space for the purpose at hand. This was Einstein's idea of his Equivalence Principle. This translates to allowing an inertial frame to approximate some portion of a situation, but only over a limited spatial extent and only for a brief time interval. Exactly how much is 'some', 'limited' and 'brief' becomes, in the mathematics, a process generically called limiting arguments.

Try this for GR : call each little part of spacetime a 'box'. These are four dimensional. Each box we will model as being inertial ie. so that we may use SR within them in all it's glory eg. describing paths of objects etc. Adjacent boxes will differ in that they maybe considered as inertial frames with a relative motion. So we may look from one box to it's neighbour and use the SR transforms that demonstrate those oft quoted length contractions and time dilations. That means we have a way of describing the same phenomena from two slightly different points of view.

Perhaps you might imagine a light ray for instance that has a 'world line' or path in spacetime, or a material body winding it's way through those four dimensions. This is a global view for which a single inertial frame will not suffice. These objects travel along their world lines. Or if you like the world line is the object. Each such curve is composed of tiny boxes strung together like beads on a string. For a given actual curve segment we might divide it up into a certain number of boxes and we can consider pairs of adjacent boxes. If we increase the number of divisions ( more but smaller beads ) we make more boxes and pairs thereof. If we increase the number of boxes in such a way as to make them all become closer and closer together, then the smaller is the deemed relative velocity between each pair and the more gradual the SR transform undergone between each member of a pair.

Now we want to divide the path into an infinite number of infinitesimal boxes. Each of these boxes will get the fancy title of Momentarily Co-Moving Reference Frame ( MCRF ), and a path will be the concatentation of such. Here is where ordinary language will likely/quickly fail us and one has to transition to the more precise formalism of mathematics. To be exact the topic of differential calculus with an overlay of tensor notation.

The differential calculus deals with those 'rates of change' or transforms between the wee boxes. Determining a rate of change is called differentiation, and is much like deciding what gear you have your car running in at some moment. There is a ratio b/w engine speed and road speed that a specific choice of gearing decides. Thus when one applies the SR transform for length contraction ( Lorentz ), that is a rate of change ie. what ( mutual frame velocity dependent ) number does one apply to correctly describe such length alteration.

{ There is a cheeky aside to this. We can say the object is unchanged but the ruler varies. This is the essence of the metric concept and underlies much of the commentary about bending/curvature in space and/or time. }

The tensor notation is one way of managing the complexity of the mathematical expressions. The Lorentz formulae in their most general form have each of the x, y, z and t coordinates in one frame depend upon all of the x, y, z and t coordinates of another frame. One could organise this complexity in, say, a matrix form. But that has some unhelpful limitations I won't go into. The tensor forms have a snappy brevity which requires not inconsiderable effort to understand ( I am by no means fully conversant in this ), but does correctly condense all that matters in GR.

Suffice to say that there is an exact methodology to precisely track all those little changes of metric as we move from little spacetime box to infinitesimally adjacent spacetime box. GR is one of those areas of knowledge where the principles are quite easy to state, not too onerous to speak of in generality, but pretty appalling ( for the uninitiated ) to handle in concrete instances.

Cheers, Mike.

( edit ) You can rewind the thinking the other way : GR should revert to SR as gravity diminishes. The wee relative frame velocities go to zero, the MCRFs become equal to each other, and you return to a single inertial frame ....

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## Now the deeper gag, the real

)

Now the deeper gag, the real deal as it were, is that nature behaves the way that The Relativities indicate even if we never set up all those clocks and rods. No known exceptions so far, despite ample testing. The 'failures' if any are due to limited misunderstanding/misapplication of the theory ie. using cake mix when you have a recipe for rabbit stew. You won't even get Rabbit Cake ! :-)

The clocks and rods construct is a way through the issues of 'what do you mean by length ?', 'what do you mean by time ?' and other geometric stuff eg. 'what do you mean by a right angle ?' Thus as you view some object descending toward a black hole's event horizon it will :

- never seem to quite get there,

- the light from it gets dimmer and redder in frequency,

- any regular motions it had seem to go slow-mo.

All of which could be summarised by saying that a clock travelling with the object appears to run slow by distant viewers. Or simply throw away the clock and just say time goes slower. Plus we also get around the awkward anthropocentric stuff ( cue solipsism ) that insists upon human presence for things to be real. Remember the use of the word 'observer' to represent the entirety of a coordinated data collection system.

{ Did the Universe not exist before the animal lineage now labelled as 'human' became self aware ? The whole cosmic background radiation field just happens to replicate the effect of a prior timeline before we became 'clever' ? Thus far at least SR/GR has a 'locality rule', meaning that I have to wait for something to propagate for cause and effect to occur. It may be that a supernova has occurred near us in the Universe but the light hasn't reached us yet. We could all be dead by tonite due to the radiation to be received one hour after I post this message. Who can say ? }

Finally : back to the observer on the planet Earth, with his gravitational force-vector meter, while looking at the Sun in the sky. Does the meter always point to the Sun ? Yes it does ( ignoring atmospheric refraction, a local irrelevance, or just go to the Moon instead ). Can you predict the course of the Earth from any chosen moment onwards by using the meter or the visual reference ? Yes, use either and the answer is the same. Can you likewise post-dict the current path/position of the Earth by using either ? Yes. Will there be a different apparent direction of the Sun in the future, with corresponding force components, due to evolution of the solar system as per known dynamics ? Yes. Etc.

GR achieves a more than adequate explanation of this by being an extrapolation of SR, which in turn factors in the light time delay from the get-go. That accounting occurs at the low level of data definition ( ... the time of an event is of the clock at the event etc ....). GR tags and transforms all the little adjustments required to track how measurements change across some system. The metric used is the one discovered that allows such transforms to be useful. It is a generalisation of the Minkowski metric.

Cheers, Mike.

( edit ) "Rene Des Cartes" : LOL! His middle name was Des ? :-)

( edit ) Any questions at all ? NB with the new web site there is no 'read' counter for posts and thus I could be talking to myself ... ah well, I could go all Matrixey about that. :-)

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## Where'd Who Go ?A common

)

Where'd Who Go ?

A common way to analyse the geometry of spacetime is to look further at the 'interval' :

Which, by the way it is defined here, will be a positive number if the distance components are greater than the time component, a negative number if the time component is greater than the distance components, and of course zero if the components cancel out. These are classified as space-like, time-like and light-like ( or null ) intervals respectively.

Thus a space-like interval b/w two events has more distance than can be crossed by a photon in the time available.

A time-like interval has more than enough time for a photon to travel b/w the two points.

Light-like/null interval has exactly the right separation in time and space for a photon to travel between.

In this language then, we were using null intervals to synchronise the clocks within an inertial frame, or if you like we made them that way by suitable clocks settings.

{ For the diagrams that follow I will suppress two of the distance axes of spacetime. Thus time will increase going upwards and the remaining spatial dimension is left-right. }

Now if you think of Newtonian spacetime ( light speed being effectively infinite ) then for a given observer at some moment, all of spacetime is divided into the 'past', the 'now' and the 'future' :

.... with 'past' and 'future' implying causation. I can affect the future ( anywhere in the cosmos after 'now' ) and the past ( anywhere in the cosmos before 'now' ) may affect me. But for Special Relativity we have :

Remembering that as time is that which is measured by a clock at an event, then 'now' is merely 'here & now'. The diagonal lines are light-like ie. photon world-lines intersecting at the 'here & now'. Future and past have the same meanings as in Newtonian spacetime, but there is a new category called 'elsewhere'.

Elsewhere is all of those events in spacetime that are neither in your past nor in your future. So over at Proxima Centauri things happen that we don't know about. Yet. In time we will be aware of them. Likewise we have things happening here & now, any effect of which will take time to get to Proxima. The duration of the elsewhere can be quite long, and more so the farther away in distance they are from our here & now :

If Event_A and Event_B are at Proxima then the duration b/w is nearly 9 years, if they are at the Andromeda Galaxy then the duration of elsewhere is about 4 million years. For our Sun there is some 16 minutes of elsewhere for each of our 'nows' here on Earth. If it's across my office then it's a handful of nanoseconds.

A famous example of elsewheres ( plural ) is the cosmic microwave background. Today this is reaching us from an entire universe's lifetime - approx 13.6 billion years - and corresponding distance away. We see photons as they were formed about 300,000 years after the Big Bang, but they've lost alot of energy ( about a thousand-fold ) in the expansion of the universe.

If you were Master Of The Universe and progressively increased the speed of light then SR spacetime would gradually become Newtonian :

.... in the limit of infinite light speed the past & future light lines become one, the 'now' ( across all of space ). An increase in the speed of light allows one to be affected by more events ( that were elsewheres and have become your past ) or to affect more events ( that were elsewheres and have become your future ).

{ Diagrams which only suppress one of the distance axes will give a conical shape that demonstrates the totality of light-like world lines. Hence the phrases 'past light cone' and 'future light cone'. Alas I continue to fail to draw 4D pictures ..... }

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## Mike Hewson wrote:[A famous

)

Mike Hewson wrote:Where did the energy go?

## AgentB wrote:Mike Hewson

)

AgentB wrote:SHORT ANSWER : Into separation of everything from everything else ie. gravitational potential energy which would be regained as kinetic if everything fell back together again. So the photons have lost energy emerging out of the common gravity well. In a sense the expansion of the universe is the ultimate 'stone throw'. How 'high' will it go ?

LONG ANSWER :While the above is true as far as we can determine for what we have observed, what adds to the confusion is where one places the baseline for energy accounting. Which in turn then relies on when/where a given theory attempts to fill in the blank space : <?!*WHAT'S IT ALL ABOUT THEN, WHERE DID THE UNIVERSE COME FROM*!?>. In the general literature and many pop science programs that substrate for discussion is frequently not plainly stated. By that I mean what/where is the entire definition of the thing we are describing ? Is our 'universe' part of another ? Is it all there is ? Are there multiple universes, some connected to ours and some not ? Etc.

One of the cuter theories is where the Universe is created by the collision of two higher dimensional objects called 'branes'. The energy to kick off the expansion is the energy in the branes. Such collisions, to those who are in the region of intersection ie. us, will look exactly like a Big Bang. So we are the products of an accident between higher dimensional vehicles and our energy is a converted form of what existed, 'kinetic' if you like, in those branes wandering about in some hyper-spatial whatever. That sounds all well and good but it isn't. You see I could follow Douglas Adams and say that the Universe was sneezed out of the nose of The Great Green Arkleseizure ( TGGA ). So we are the snot from TGGA's sinuses, and perchance if you were in that mucus it would all look like the Big Bang too. However neither has a specific observable signature that distinguishes them. So in a logical/semantic sense you can interchange one for the other and still see a cosmic microwave background etc. By their very construction the Braneworld and TGGA explanations are bound to agree with what we already know to be observably true. Neither are a physical theory of interest because neither proposes an observable phenomenon to test it's truth by.

The deeper problem here is that GR only dictates how universes change. It doesn't prescribe any specific states at any point. It just says : given this state A, then it will evolve to this other state B in such & such a manner.

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal