Rudimentary Relativity with Loedel Diagrams

Part I

  • Source: The spacetime world, loedel’s spacetime diagram: Applications of physics (UMAP modules in undergraduate mathematics and its application), Publisher: ‎ COMAP, Inc. (January 1, 1993).
    • “Spacetime diagrams of various types provide geometric constructions equivalent to the Lorentz transformation between uniformly moving reference frames. The best-known and one of the oldest spacetime diagrams is the Minkowski construction… Although the Minkowski diagram is particularly useful for treating such problems as the Twin Paradox, it is complicated by the necessity of using different scales along the two sets of spacetime axes. A much simpler geometric method was presented in 1948 by Enrique Loedel. [Loedel’s paper, Aberración Relatividad, published in Anales de la Cientifica Argentina 145 (Jan. 1948), attracted little attention in the scientific community until the same method was independently presented by Henri Amar in the American Journal of Physics 23, 487 (Nov. ’55).]”
    • “… unprimed X and CT axes represent positions and times as determined by an observer O, while axes labeled X’ and CT’ represent the corresponding spacetime coordinates as determined by O’. This second observer is fixed at the origin of an x’y’z’ space reference frame which moves uniformly along the common direction of increasing X and X’.”
    • “In Loedel’s spacetime diagram, both sets of axes are tilted. This feature, which allows primed and unprimed axes of the spacetime diagram to be drawn to the same scale, is the principal advantage of this type of diagram. Figure 2 shows how the axes of the Loedel diagram are arranged.
    • A worldline traces an observer’s path through spacetime. The CT and CT’ axes are the worldlines of two observers: O and O’.
    • Lines of simultaneity connect all simultaneous events and establish chronological order in a given reference frame.
    • A Loedel diagram preserves the essential content of Einstein’s theory of special relativity.
    • In the next diagram, the vertical CT:CT’ and the perpendicular, horizontal X:X’ lines represent two observers at rest with respect to each other.

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Relativity is, of course, no fun if you don’t use it to tell stories. Loedel diagrams make some stories told with relativity really, really good stories.

My favorite is one that I came up with myself. Initially, I started out with two rocket ships, one blue and one red, and two sets of identical triplets, one set in each rocket ship. I began the story at the most exciting part, right when the ships–flying in space toward each other–passed each other and flew away from each other, all in opposite directions. For special effects, I decided that the ships and their triplet occupants would travel at 0.8660c with respect to each other. That enabled me to tilt the blue ship triplets’ worldlines a noticeble 60 degrees relative to the red ship triplets’ worldlines, like this:

The fact that I put six triplets in the ships and noted who waved at who when allowed me to identify nine idealized events and record their chronological order from a blue ship triplets’ point of view and again, from a red ship triplets’ point of view.

Altogether, the tripletts’ worldlines, the events’, and the lines of simultaneity from two perspectives, make for a pretty picture:

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As fascinating as all of that is, I am even more impressed by this heavenly view of the foregoing "thought experiment:


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I have always liked this classic thought experiment (i.e. story):

explained here:

The short of it, the person on the train moving at 0.5c observes the light particle moving half as far over the same time period as that witnessed by someone standing outside the train.

What does that look like for your intersecting lines of simultaneity?


Couldn’t tell you right off hand; there’s the slightest chance that I may be able to figure it out, but that would have to wait until I finish playing with my set of Loedel diagrams, unless you wanted to try your hand at drawing some yourself and show me in this thread.

And there’s more…

it occurred to me that there’s a possible scenario in which the two ships begin at rest with respect to each other, pass each other, and end at rest with respect to each other again.
This is what I “see”:

Continuing from my assertion in Post #7, I would be pleasantly surprised if a reader of that post were able to figure out the separate paths of the red and blue rocket ships and agreed with me. But–for the benefit of those who aren’t so quick in figuring out the ships’ paths–the following two figures chart the paths of both ships, IMO.

Both ships leave their initial positions, accelerating in opposite directions, until they pass each other at the half-way point of their separate journeys, and continue on, eventually coming to rest with respect to each other. That is, IMO, quintessential relative motion of any two objects in the cosmos.

Which leads me to a question that I have asked and has never been satisfactorily answered.
I was informed, by a knowledgeable and competent relativist, that the correct answer to the question: “Which twin, in the Twin Paradox is younger?” is: “The one with the shortest worldline.”

If true, my question is: Who’s the youngest of the triplets in my "Double Triplet Scenario?

IMO, Relativity has been misunderstood and abused to fill gaps in all kinds of areas of modern physics. The primary source of confusion is a lack of awareness that relativity exists in the RECEIVED DATA and not in the objects under observation themselves.

Relativity is necessarily true because light travels at a finite speed. Objects moving at significant percentages of the speed of light are either speeding away from their own signals (thus increasing the apparent spacing between them), or catching up with their own signals (thus decreasing the apparent spacing between them). That’s why the mathematical transforms are necessary; they mathematically account for the discrepancies in the received data that has been skewed (dilated or compressed) by the speed and/or distance the signal has traveled. But in no case is the distortion in the actual observed object itself. No object is physically being foreshortened or lengthened; local time is not changing for either party; it is only an apparent distortion in the data.

If the observed phenomenon were the absolute reality, there would be no need for the mathematical transforms. The picture the received data shows would be the reality. The ‘worldview’ or ‘God’ views of diagrams may be interesting from a thought experiment POV, but they in no way reflect the reality of what the players or objects in the scenario would actually observe, and therefore add nothing to comparing observations between the objects themselves.

In regard to the twin paradox, it isn’t a paradox at all; it’s an incorrect formulation of relativity. There is no younger or older in the scenario; the passage of time is equivalent and absolute for both twins. If ‘local’ time for each twin were different, there would be no basis of comparison of the observed times between the two. They have to agree on what local time is before they can say that what they observe doesn’t agree with local time. You can’t compare measured lengths of an object if the parties don’t agree with what an actual ‘inch’ is. Further, the statement of the Twin Paradox has an error; it is stated that the stationary twin sees time dilated in regard to the travelling twin in BOTH directions. This is false; dilation would only be observed as the travelling twin is moving away. Once the return trip begins, the stationary twin will observe time compression. And the amount of dilation and compression will be equivalent such that the twins will be the exact same age once they join up again. Again absolute time has not changed and passes at the same rate for both twins; it is only the received data that the stationary twin gets that displays the discrepancy, the discrepancy is not the reality of either party.

That’s a remarkable claim and suggests to me that you may not be aware of the Paradox of the Light Sphere".

Unfortunately, I do not know everything about everything AND I am as well aware of some things as lacking in awareness of many other things. However, I think if you were to follow through my “Double Triplet Scenario”, the one thing that you would find is that I am very, very aware of the fact that evidence

  • "… must be interpreted and it is always interpreted either on the basis of some theory or else on the basis of some set of inchoate presumptions. If one starts out with false fundamental notions, then one will also read falsity that is not in the evidence itself into the interpretation of that evidence. If the evidence seems not to make sense, that is your clue that an examination of the assumptions used in interpreting that evidence is in order.”
  • In other words, I am well aware that the “DATA” that you speak of is “RECEIVED” and “INTERPRETED” "on the basis of some theory or else on the basis of some set of inchoate presumptions".

Incorrect. It is because the speed of light in a vacuum is the same for all inertial reference frames. For that reason, it actually is more like an infinite speed which only looks finite to observers. If you try to race light to a destination, it will continue to race ahead of you the same no matter how fast you go. And you can get to the destination as fast as you would like. It is frankly more about the structure of space-time than about light.

Incorrect. That is an entirely different phenomenon called the aberration of light. And it is weird effect on its own, where the faster you go the farther away your destination appears to be, even though it is actually closer because of Lorentz contraction in relativity.

How do you explain the Hafele-Keating experiment?


It’s always been a personal pleasure to watch an uninformed relativist get schooled in relativity.

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My previous mention of The Paradox of Light Sphere leads me to post more information about it.

  • In Einstein’s 1905 paper, ON THE ELECTRODYNAMICS OF MOVING BODIES, he wrote: “We …also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.”
  • That claim–the second of two postulates in Einstein’s theory of special relativity–is, inspite of its wording, a remarkable claim which I am told is the basis for the non-paradoxical Paradox of the Light Sphere
  • Because my experience has been that the quickest way to loss of credibility when talking about “the theory of special relativity” is to paraphrase or interpret any aspect of it inaccurately, I prefer to quote “established and acceptable authorities” excessively and let others bungle the interpretation. Regarding the “Light Sphere” claim, I cite and/or quote two sources.
  • My second source is a citation and quote from well-known and deceased non-mainstream scientist, Harry Hamilton Ricker III: “The Light Sphere Paradox Of Special Relativity”( › papers_download)
    • The best introductory discussion of the paradox is presented in Basic Concepts of Relativity, by R. H. Good, pages 39 and 40. Good asks the reader “Suppose a light pulse emitted from a stationary source at the origin at time t=0. It then spreads out with velocity c in a spherical wave front, as seen by an observer at rest relative to the source…Now an observer in another frame of reference moving uniformly in the x direction with velocity v relative to the first frame will also see a spherical wave front spreading out with the velocity c from the origin of his reference frame, assuming that the origins coincide at t=0. This is not what one would expect intuitively; one would expect the sphere of light to remain centered on the source, and consequently to move with different velocities in different directions away from the origin, as seen by the observer in the moving frame.

In this post, I’m going to step out on a limb and offer a Loedel diagram that illustrates the Paradox of the Light Sphere described in my last post. “Stepping out on a limb” merely says that I think the diagram is accurate, but I’ve never shown it anyone. So, I don’t have independent, reputable, authority that confirms that my diagram is accurate. The context of the diagram is my Double Triplet Scenario, specifically the two ships’ journeys past each other.

In this post, assume that a light source located between the two moving ships emits a single flash of light into the fronts of both ships at the same instant; a flash which expands from the front of each ship to the back of each ship. IMO, the following Loedel diagram describes the consequence of the flash in and with respect to the two reference frames in motion at 0,8660c relative to each other.

I’m working on an answer now. It looks kind of challenging, but I have a suspicion that, if and when I find the answer, it’s going to be “plain as the nose on my face.” Coming to a thread near you.

Those are my favorite type of solutions. Thanks for giving it a go!

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  • I deleted my first and second efforts.

  • Figure 4a. is a brief summary of essential equations for those prefer calculating numbers.

  • Figure 4b is a diagram of two views of a light’s round-trip jouney in a lightclock: one view from the point of view of an observer at rest in the train, and the other view from the point of view of an observer at rest outside the train.

@T_Aquaticus , Ha! back to the drawing board. I found errors.

My third and final effort. If it isn’t correct, I give up. Don’t ask me for calculations, I do pictures, not calculations.

  • Loedel version of the Relativity’s Light Clock
  • Opening Figure #1, barebones leftovers from my final ***Paradox of the Light Sphere" diagram.

  • Figure #2. First leg of the light beam’s journey from its source to the other side of the train.

  • Figure 3. Four destination alternatives, lettered “A.”, “B.”, “C.”, and “D.”

  • Figure 4. Four destination alternatives, with letters and arrows.

  • Figure 5. Second and final leg of the light beam’s round-trip journey to Destination D.

@T_aquaticus , I’m done.

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