But trying to show how I found apparently no obvious errors in his basic thought experiment was not as clear as when I first figured it out so this version a bit rough:
Please allow me to try and follow the thought experiment of Dr. Dick.
A Thought Experiment for those who like to think!
Consider any experiment which can be performed (if your math abilities are limited, I suggest you make it a fairly simple experiment). Now make an analytical examination of that experiment from any standard space-time frame of reference you find convenient."
*****I propose this experiment: two balls moving in space collide into each other and move apart again.
Graph of experiment: I put "time" as vertical axis. I put "space" as horizontal axis.
I show the two balls at the base of the graph with a gap between them, both balls on the horizontal axis at zero time. Above here I show the balls are now closer together at one unit time. Looking further above I have the balls closer and closer for higher values of time till they are together, then further up they progressively move apart as climb higher up the time axis.
I also draw another graph representing the view from on one of the balls. The left ball is seen on the graph to describe a straight vertical line parallel to the vertical time axis.
The right ball is seen to approach it from a bigger distance than in my first graph; to meet it part way up the time-line of the left-ball whose view I'm adopting here, then move away again. Note in the first graph I got equal curved paths for each ball, in the second graph I got a straight line for one ball and a deeper curve for the other.
Just two ways of looking at two balls approaching, colliding, moving apart; on a space-time graph.
"That is, solve the problem and analytically represent the relativistic correct solution. There is no argument with Relativity here at all! If you can not see that, you do not understand Relativity."
*****Doesn't seem to be a problem?
No matter what that experiment entails, it can be seen as a number of things (particles, objects or collections of such) which travel along trajectories in that space-time frame of reference you chose. Let us not worry, for the moment, where those trajectories start or finish but rather just choose some arbitrary start point on each path and finish with some arbitrary stop point on the same trajectory."
Again, there is nothing here in contradiction to any Physics taught at any University in the world and anyone competent in Physics would know that. We are doing nothing but making a careful examination of the experiment you the reader have proposed."
*****I can take an arbitrary location of start and finish for the left ball and for the right ball I guess.
"quote: Go on with that careful analysis!
Now, those trajectories are lines (in Einstein's space-time continuum) and they may be discussed in terms of a parameter along their length say for example "p". It follow that any given line in that experiment may be described by the events which constitute that line: for any given p, the space-time coordinates correspond to the collection of events (x,y,z,t)p .
These can be seen as continuos functions of p or as a tabulated selection of a finite number of events; either perspective is a reasonable representation of the space-time path of the thing of interest. Do this for each "thing" involved in your experiment.
If representing lines in a geometry through a parameteric notation is beyond your experience, your understanding of mathematics is far to limited to understand the above step and you will have to drop out of this discussion. If you have any competence in Physics, you should find no difficulty with that step."
*****"p" appears to designate a confining definition of the object. For example if the right ball had previously bumped into something else it would carry the effect of this with it so that pre-set defining characteristic of the right ball could be called "p".
A line on my graph can be described by the events that constitute that line?
For example a line of changing position for a ball leaving out the collision would be one sample of the ball's track.
Another line that included the collision would be another sample of the ball's track.
Any pre-set defining characteristic of the ball; that is any "p", the space-time co-ordinates correspond to (x,y,z,t)p? Seems O.K.?
So now, use that parametric representation!
Now, let us examine the paths of those entities of interest, each by itself, in the absence of the others (I am presuming you know enough physics to solve the problem expressed in your experiment). The differential path length along the trajectory is exactly what is referred to as Einstein's invariant interval along the path of the thing being represented by that path. This fact is commonly used in high energy physics to determine the expected apparent path lengths (in x,y,z space) of particles with short half lives.
Anyone competent in Physics already knows that so, if you find the statement confusing, you just are not sufficiently knowledgeable in Physics to follow what I am talking about. In particular, I again say there is nothing here which is not perfectly understood physics by anyone decently trained in the field."
*****Looking say at one ball: if there was no pre-set defining characteristic to the ball (no "p") then on my space-time graph paper its trajectory has a length which just defines that it is moving according to my choice of units of space and of time? The length is the "invariant interval" as it just defines the invariance of my graph locking together of space unit and time unit?
If there is a pre-set defining characteristic "p" then the ball path length will be biased in my chosen space and time units according to how this pre-set characteristic constantly distorts my chosen units?
This is here only to remind those who need reminding!
If you understood the common uses of relativity, you would understand that, in order to make the physics independent of the frame of reference, the half life (or any other temporal phenomena defined by the laws of physics, clocks included) will always be the same if measured in the rest frame of the thing of interest.
The associated points of interest along the space-time path are obtained by integrating Einstein's invariant interval along that path. Since there is no movement of the entity in its own rest frame, the interval in this case is always imaginary: i.e., the interval is time like. As a consequence, one usually uses the variable tau (via ic tau) to represent such a variable.
By the way, is there anyone out there who really believes that a clock attached to the entity whose path we are following would not read exactly the invariant interval along the path? If so, you need to go back to school."
*****The whole path is invariant interval, so just gives the TIME AXIS of the graph. If no movement of the entity in its own rest frame, you get the time axis (so time is imaginary). If tau represents this alleged variable: TAU IS THE TIME AXIS.
In my graph of two balls that approach, collide, and move apart: any path length on the time-vertical-axis, space-horizontal-axis, graph; requires a change in position on the time axis (except teleportation). If the time-axis is regarded as fixed as in a graph and made of fixed invariant intervals; the length of an object's path on the graph will be locked on to that defining-graph? If the time-axis only has one unit from start to finish, then any path length will just be one interval of time as defined by the time axis.
If you differentiate out the time-axis component of a path of a ball, on a fixed-divisions time-axis graph; the length you get is just the time-axis defined view of the ball's movement?
So the ball as clock? The ball projecting its definition of movement on to the time axis, so to speak, or vice versa?
A clock attached to the entity whose path we are following would read exactly the invariant interval along the path? By definition isn't that what a clock may be? We have a graph-clock called the time-axis; and a local clock attached to the item? Both are defined as fixed spaces or fixed rates of change with a mutual fixed space or fixed rate of change between them?
(Sounds like "offer wave:confirmation wave" deal on fixed spaces.) A clock is supposed to produce self-refering references (like pendulum allegedly retrace its path). The path is defined by the graph-clock (the time axis) so a local clock on the path will be also indirectly referring to graph-clock?
"We now have all the information we need to explicitly examine the exact nature of those trajectories, including, by the way, the nature of the original geometry itself through the explicit relationship between all of the various tau p defined by the various trajectories defined by (x,y,z)p
(through the fact that we know that tau constitutes the integral of the metric).
quote: A summary of what we now have!
At this point, if you have a strong enough math background, it should be clear that you can specify a collection of numbers (x,y,z,t,tau)p
for each and every trajectory in that experiment we were describing (let each of those lines begin with a specification tau = 0 then an integral will specify the rest).
Now, from the standard relativistic perspective, one uses a geometry of x, y, z and t with a metric which yields tau as the path length. That will require exactly the standard Minkowski geometry (if you don't use Minkowski geometry, you will not get the tau we just specified in that parameterized representation of the path).
I hope you will agree that the collection of parameterized paths of all the objects in your experiment exactly describe the experiment: by exact, I mean the correct result as deduced by modern physics.
Is this true or false; is it in anyway a misrepresentation of Modern Physics as taught in any school in the World?"
*****When I first ran through your argument I found no errors as far as you went except at the end I felt one could start with all sorts of geometries (not just Euclidean) and get them projected back again. Trying to write out how I followed the arguments is not as easy as when I worked it out. As far as it goes you seemed to be correct in saying that you put forth no theory or anything new other than a potentially interesting perspective on what is already taught.
The various "tau" I suggest are various ways of looking at the time axis on my graph.
We know that tau constitutes the integral that is common ground of the metric. IN (x,y,z)p I see "p" as rule governing how time-axis will appear.
(x,y,z,t,tau)p looks like this to me: "tau" is a difference in perspective on the time-axis on my graph; "p" is a rule for making the time-axis look different.
If you let each line begin with tau = 0 you are saying the only difference in their views of the time axis are the "initial conditions" that is the pre-set differences between the items so their mutual definition of each other say?
By setting tau = 0 you get like a Bose-Einstein condensate as the items are frozen in mutual definition or counting of each other?
You have collapsed offer waves and confirmation waves at the beginning from John Cramer's Transactional Interpretation of qm perspective?
On my space-time graph of two balls moving toward each other, colliding, and moving apart; "p" was a rule affecting (x,y,z,t). "p" was like "speed" or "acceleration" or "deceleration". A continuous "p" was equal spaces or rate of change in spaces along a path; a broken p allows interactions along a path (sudden changes in spaces?).
"p" denotes a "clock as the whole path" (?) or a local clock as a changing path.
"p" gives variation in the path. If object is accelerating/ decelerating "p" gives rate of change. If the ball on my graph coming in from the right was accelerating it would describe a longer line that swings in from the right from further away than on other graph and would be deflected less in the collision.
"p" defines the object as either locked on to "time" on the graph or a fixed "rate of change" of time on the graph so "p" is a synchronised local clock. Broken "p" gives a synchronised breaks clock (looks like Morse code!).
Regarding "standard relativistic perspective using geometry of x,y,z and t with a metric which yields tau as the path length":
"as path length" can be "as acceleration/deceleration or fixed path; affects path length.
With "time" on vertical axis and "space" on horizontal axis: a line segment on the graph is tau where it represents a change in the pre-set definition of the item relative its fixed defining relationship with the other items (as tau was set to zero for them all to start)?
(Of course if you are going to look at a collection of items from the point of view of a fixed defining relationship between them at the start; any mapping of the logically consistent changes in these items on a graph with axises in any dimensions you want (not just labelled as "tau" or "time") will be a mapping of changes in these items!?
And you can look at a particular detail, compare it with the rest of the items, and from the point of view of requiring they be seen through the perspective of their fixed initial defining relationships: you will get a way the objects have of defining your geometry, your bunch of assorted axises in various dimensions?)
If you examine an individual path in the absence of the others you get invariant interval along path? Obviously as "p" defines a constant atribute of the path (or fixed collection of atributes distributed over path in a fixed way)?
If "p" is a pre-set rule covering a defining characteristic of the item then the rate of change of "p" with respect to the time axis will by definition give an invariant interval (or fixed scale for time axis as this axis is being seen as locked to a conserved "p" rule?)
"quote: Now, the strange perspective I would like to discuss!
Now let us look at an alternate perspective of exactly the same parameterized paths. This time let us use a geometry consisting of x,y,z and tau with a metric which yields t as the path length. If you are able to do the math, you will notice that this will require exactly a Euclidean geometry (if you don't use a Euclidean geometry, you will not get the t we just specified in that parameterized representation of the path).
Once again, the collection of parameterized paths of all the objects in you experiment exactly describe the experiment: and once again, by exact, I mean the correct result as deduced by modern physics.
Once again, I state that there is no new physics here at all. There is nothing here but a rather strange perspective which I doubt anyone has ever taken and I would like to discuss the consequences with someone competent in Modern Physics.
If, in the over four thousand members, there is no one out there with sufficient training to follow that rather streight forward thought experiment, I will not bother anyone at this forum again as it is clearly a waste of time. If this post is removed from the "General Physics" forum, I will take it as a sign that the people in charge are not interested in physics."
****When I followed your post to here it seemed O.K. with no obvious errors on the face of it; just pure logical deduction with in a way no new physics just a way of looking at it. But I made a mess of trying to recover my understanding and notes into this reply in limited time so what I wrote here may be only partly clear.
When you proposed a geometry with x,y,z and tau with a metric that yields t as the path length it was not that surprising to me as by then your argument already showed that "tau" is a way of looking at the time-axis so the two were interchangeable as it was a case of juggling pre-set mutual definitions and "the space-time graph"?
This graph I had in Euclidean geometry was a way of making up a picture of inter-dependent changing definitions that looked a particular way in that perspective?
My vertical axis -time; horizontal axis -space; graph with tau as change in local definition of item relative the other items from a fixed definition-lock (?): now becomes a graph with vertical tau axis and horizontal space axis with time as change in definition of item relative the other items from a fixed definition-lock.
A "time" line segment on the tau-space graph will be longer or shorter if the object accelerates or decelerates instead of "acceleration less time/ deceleration: more time" on a time-space graph.
"p" rules only depend on each other so no external clock (like QED, some cancel). Going where the time is least might be "going where the error "time" is least? As REAL TIME is with the law of non-contradiction! "He shall come in glory at the end of time and His Kingdom shall have no end" comes to mind.
In the space-time graph acceleration involved moving in less time; in the space-tau graph acceleration involves rapid change in "pre-set all-objects-fixed-initial-definition-referent movement of particular object" definition of object so less tau (less pre-set content in its definition, more local content?) ??? So a changing perspective on the other objects?
In space-time graph: Less tau: less different perspective on time axis; more tau as more different perspective on time axis?
If something accelerates relative something else it overtakes it; to a third object (say the background) that locks the two passing items together: the background could go with the overtaking object (making the other appear to pull back) or it could go with the passed object (making the other appear to pull away).
A sum (integral) over a path will specify the rest of the items as a holographic view of the other items?
"p" as a "blackbody that emits or absorbs defining characteristics"?
General relativity: involves curved trajectories and changing speeds. How see these when the parameter "p" looks like these? Requires a discontinuity in the path definition: a jump or gap during which there can be a sudden change in alternative way of looking at the object?
Peter Lynds notes that relative motion means position cannot be precisely determined. Actually by definition "relative" locks two items together in mutual definition of motion? The background could be going partly with either object, or locked on to either object?
"Do clocks measure time?" looks similar question: graph-time is like "third-party-time" or "background-time"; if a clock requires movement to operate then this movement may be partly coloured by background movement?
Seems possible to see why Lee Smolin came up with quantum geometry as can see that geometry could change with each event but fixing a geometry would require discontinuities?
I know this is very messy post.
Not so sure if I agree as much as before.