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Thread: GR without spacetime manifold

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    GR without spacetime manifold

    Can you imagine general relativity without a continuous spacetime manifold? Both concepts seem to be inseparable, but I will show here that Lorentzian manifolds are even not compatible with GR. The principles of GR are describing worldlines, but not the vacuum points between worldlines.

    After the development of Einstein's special relativity, Minkowski presented in 1908 his model of a continuous spacetime manifold. Later it was Einstein himself who introduced a curved spacetime manifold for the mathematical description of gravity. This mathematical model of a curved spacetime manifold was considered an inseparable part of general relativity, but to date it has not been possible to harmonize it with quantum mechanics.

    In the following it will be shown that the concept of a Lorentzian manifold does not only not comply with quantum mechanics, but that it is even not compatible with the principles of GR itself, and that spacetime is limited to worldlines, at the exclusion of vacuum points between worldlines, because
    I. Vacuum is not defined,
    II. There is no coherent interval concept for Lorentzian spacetime,
    III. Coordinates of observers are continuous manifolds, but they have no Lorentzian metric.

    I. Vacuum is not defined by GR
    The issue is shown very clearly by special relativity, with its two postulates:
    - The laws of physics are the same in all inertial reference frames.
    - Speed of light is measured with the same value c in all inertial reference frames.

    Manifestly, both postulates are talking about inertial reference frames (and also about lightlike phenomena), but not about the vacuum between worldlines. In particular, time evolution is assigned to worldlines, but it is not defined for vacuum which seems to be a timeless void. Vacuum is defined by quantum physics (and possibly by cosmology with respect to dark energy), but neither by SR nor by the gravitational spacetime curvature models of GR.

    II. GR has no coherent spacetime interval concept - spacelike intervals would imply square roots of negative numbers

    Example: A spaceship is traveling from A to B in 5 years, the space distance between A and B is 3 light years - what is the spacetime interval between A and B?
    Physics provides 4 contradicting answers: the spacetime interval could be 4, 4i, 16 or -16, depending on our choice of signature (+,-,-,-) or (-,+,+,+), and of extraction of the root or not. But whatever is our choice, we get imaginary spacetime distances (or negative squares) in one sense, either for timelike or for spacelike intervals.

    This dilemma is due to an intrinsic deficiency of Lorentzian "manifolds" whose definition is limited to worldlines, at the exclusion of vacuum points between worldlines, and this is why spacelike spacetime intervals are imaginary and meaningless.

    III. Coordinates of observers are continuous manifolds, but they have no Lorentzian metric.

    What about the coordinates of observers? They seem to be continuous manifolds, and vacuum points between worldlines seem to be defined. The answer is yes, they are manifolds, but they are no pseudo-Riemannian manifolds. As soon as we try to introduce a Lorentzian metric onto the coordinates, the continuity get's lost, only worldlines are remaining, at the exclusion of vacuum points between worldlines, and only worldlines are transforming from one observer to another.

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    Quote Originally Posted by Rinaldo View Post
    Can you imagine general relativity without a continuous spacetime manifold?....
    The answer is no because that is not GR, Rinaldo! This is General relativity.
    Your errors.
    1. Vacuum had a definition before SR or GR existed.
      GR uses the textbook definition of vacuum, e.g. see the vacuum solutions of GR.
    2. A "the vacuum between worldlines" error.
      A world line is a mathematical line of an object traveling through a mathematical spacetime. No vacuum (or air or soil or plasma, etc.) is involved because a world line is not a physical line.
    3. GR has a coherent spacetime interval concept.
      GR has the coherent concept of the metric tensor. The GR concept of a spacetime interval is based on the metric.
      A badly formed example.
      • The spacetime interval between your two spacecraft depends on the metric being used and their speed and the gravitational field they are in. The example has no metric or speeds or gravity !
      • "4, 4i, 16 or -16" numbers appearing out of nowhere.
    4. A "Lorentzian "manifolds" whose definition is limited to worldlines" error.
      Manifold. A manifold is a specific topological space that is never limited to world lines. World lines are the paths of objects in manifolds.
    5. Coordinates of observers are continuous Lorentzian manifolds in GR.
      A special case used in general relativity [of a pseudo-Riemannian manifold] is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike. A Lorentzian manifold is a manifold with a Lorentzian metric.
    Last edited by Reality Check; 2019-Dec-06 at 12:09 AM.

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    Quote Originally Posted by Reality Check View Post
    1. Vacuum had a definition before SR or GR existed.
    GR uses the textbook definition of vacuum, e.g. see the vacuum solutions of GR
    Reality Check,

    Vacuum, according to Wikipedia, is "space devoid of matter." The key question here is if such void (vacuum) is able to provide a manifold with continuity. This is true for the 3D space manifold, and Mainstream assumes that this is also true for 4D Lorentzian spacetime, but I am contesting such a possibility.

    The historical definition of vacuum before SR and GR relates to 3D space, not to spacetime. Vacuum solutions are not helpful here: "In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically."(Wikipedia). Vacuum solutions do not refer to the vacuum, but to the absence of spacetime curvature.

    2. No vacuum (...) is involved
    I only refer to Mainstream's assumption of a spacetime manifold, physical or not. In this spacetime manifold there are worldlines of particles. Between the worldlines there is void. Again, the key question here is if such void (vacuum) is able to provide a manifold with continuity.

    3. A badly formed example
    For clarification, here is a completed description of the example:
    A spaceship is traveling with constant speed in outer space where gravitational effects may be neglected, from A to B. An observer observes that the spaceship, according to the observer's reference frame, is traveling 5 years, and he measures a distance of 3 light years between A and B.
    Then, based on this information, the observer calculates the spacetime interval between A and B.
    For this calculation he must choose between two signatures (+,-,-,-) and (-,+,+,+). The first one which is used in particular by particle physicists provides the equation ds2 = t2 - x2 -y2 -z2 = 25-9 = 16. The second signature is used in particular following Misner Thorne Wheeler, the equation is ds2 = -t2 + x2 +y2 +z2 = -25 + 9 = -16, yielding a negative square for the spacetime interval. Furthermore, some authors are extracting the root of ds2, and some don't. Those who are extracting the root get, always depending on the chosen signature, 4 or 4i.

    It is exactly this contradiction which is ignored by Mainstream's assumption of the continuity of spacetime. By the way, the assumption of continuity of spacetime is used in particular in quantum gravity, and the reason why quantization of spacetime cannot work is that it is based on this wrong assumption of a spacetime manifold.


    4. A manifold is a specific topological space that is never limited to worldlines
    That is correct, but I want to say: As the Lorentzian metric is only limited to worldlines, it is not possible that it forms a manifold.

    5. "Coordinates of observers are continuous Lorentzian manifolds in GR.
    I was mentioning the coordinates of observers just in order to show where we could find a continuous spacetime manifold: I am referring to coordinates such as Minkowski diagrams (the same principle applies also to coordinates of higher dimension) which are just like sheets of paper: No doubt that sheets of paper are continuous. But in Minkowski diagrams the Lorentzian metric is invisible, the distances (intervals) we are measuring with a ruler on a sheet of paper do not correspond to the spacetime intervals, and as soon as we define a Lorentzian metric for these coordinates, the continuous manifold character gets lost.

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    Quote Originally Posted by Rinaldo View Post
    Vacuum, according to Wikipedia, is "space devoid of matter."...
    You are confusing physical space with its vacuum with a mathematical space which has nothing in it except mathematical objects. A mathematical manifold is defined to be continuous. You are ignoring the mathematical definition of a manifold. That is why you are wrong.
    That error is repeated. There is no "void" between mathematical world lines. A mathematical manifold is defined to be continuous.

    A fuller example shows your error. The choice of signature is a convention. It is invalid to compare results from one convention to another. It is analogous to complaining that a result in inches is different from a result in centimeters.

    You are still confusing world lines of objects moving through spacetime with the entire manifold and its metric. Read my source again. A special case used in general relativity [of a pseudo-Riemannian manifold] is a four-dimensional Lorentzian manifoldf or modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike. A Lorentzian manifold is a manifold with a Lorentzian metric.
    This is a manifold.
    To get a Lorentzian manifold we add a Lorentzian metric. This defines the geometry between all of the points in the manifold.
    Now add an object. The path of this object through the Lorentzian manifold is a world line. That is a line through the Lorentzian manifold.

    Perhaps a formal question to think about will make this clearer.
    Think about a total empty universe (look up vacuum solutions of GR). A Lorentzian manifold exists because that is part of the mathematics of GR. Now add an object to this empty universe.
    IF01: What happens to the Lorentzian manifold according to the definition of a Lorentzian manifold, Rinaldo?

    FYI An answer is nothing in GR because the object's world line is just a 4D line in the existing Lorentzian manifold.

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    Reality Check,

    In my initial post #1 you find clearly separated the physical issue (I.) and the mathematical issue (II.) of Lorentzian manifolds. Physically, the vacuum between worldlines is not defined by relativity, and mathematically, there is no coherent interval concept for Lorentzian spacetime. These are two reasons why there can be no manifold which could fill continuously the vacuum between worldlines.

    The choice of signature is a convention

    The scientific dispute on conventions is hiding the real issue:

    The history of a hidden error


    In 1908, Minkowski presented the signature (+,-,-,-) which seems plausible, and which never received significant contestation. However, without mathematical urgency, a second signature (-,+,+,+) was introduced, leading to a double-tongued terminology. In 1973, Misner-Thorne-Wheeler published their famous work on gravitation, and they decided to opt out of Minkowski's signature. The terminology dispute was now cemented, this is particularly strange because the signature did not seem to be an important physical issue.

    Why did MTW deviate from the Minkowski signature (+,-,-,-)? Probably because of strange results for spacelike intervals which are becoming imaginary: A space distance of 5 meters corresponds to a spacetime interval of 5i meters (!). Instead of exposing this inconsistency of the Lorentzian manifold, MTW decided simply to replace the signature with the signature (-,+,+,+), hiding by this the inconsistency. But even after having changed the signature, the fundamental problem remains: 1. Timelike spacetime intervals become imaginary, 2. Proper time and the spacetime interval become two separate notions, 3. Particle physicists preferably continue to use the Minkowski signature (+,-,-,-).

    By consequence: The only Lorentzian interval is the timelike interval, in relativity this is proper time. Spacelike spacetime intervals are imaginary and lack any theoretical foundation. Mathematically, Lorentzian manifolds cannot exist.


    IF01: I contest the possibility of any (continuous) Lorentzian manifold. The effect of the mass of an object is - according to mainstream - the curvature of the spacetime metric. However, the model of curvature of spacetime by gravity is not compatible with the experimental evidence of quantum mechanics.

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    Quote Originally Posted by Rinaldo View Post
    ...
    I read the OP and have stated the fundamental reason that it is wrong: Lorentzian manifolds are compatible with GR.
    Someone's "The history of a hidden error" nonsense about scientific convention without a source. Repeating personal, unsupported beliefs is not an answer to
    IF01: What happens to the Lorentzian manifold according to the definition of a Lorentzian manifold, Rinaldo?

    In case you have more than personal, unsupported beliefs to support "Mathematically, Lorentzian manifolds cannot exist".
    IF02: Give your derivation that Lorentzian manifolds are not compatible with GR using valid math and physics, Rinaldo.
    What you have stated is that the mainstream Lorentzian manifolds are not compatible with GR even though they have been in GR for 104 years and no one has ever noticed it!
    No vacuum irrelevancy - a Lorentzian manifold is a mathematical object. No world line irrelevancy - world lines are paths in a manifold, not an entire manifold. No incompatibility between GR and QM - you made a statement about GR! No convention irrelevance - that is a convention and the results from different conventions cannot be compared.

    "However, the model of curvature of spacetime by gravity is not compatible with the experimental evidence of quantum mechanics." is irrelevant and slightly wrong. A quantum gravity theory has been extremely difficult to form because there are fundamental differences between GR and QM theories. We have no experimental evidence that GR and QM are incompatible because that would happen at scales of the Planck length. The proposed solutions generally have a curved spacetime as a cause of gravity because they reduce to GR in the appropriate limit. For example, look at string theory which has very compact extra dimensions that give QM and the same spacetime as GR at higher scales.

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    Quote Originally Posted by Rinaldo View Post
    IF01: I contest the possibility of any (continuous) Lorentzian manifold. The effect of the mass of an object is - according to mainstream - the curvature of the spacetime metric. However, the model of curvature of spacetime by gravity is not compatible with the experimental evidence of quantum mechanics.
    How is this relevant? It is well understood that gravity in GR cannot be quantised. So this contradicts your claim. Anyway, there is nothing in quantum theory that says that space or time must be quantised. And all experimental tests are inconsistent with spacetime being quantised.

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    Quote Originally Posted by Strange View Post
    How is this relevant? It is well understood that gravity in GR cannot be quantised. So this contradicts your claim.

    I do not understand, Strange. Are you talking about a particular theory of quantum gravity? Quantization of curved spacetime is still attempted by quite a lot of currents of quantum gravity.

    Anyway, there is nothing in quantum theory that says that space or time must be quantised.
    Yes, in a more general way, it is argumented that there must not necessarily be a theory of quantum gravity. You are right that this is a restriction to my argument about inconsistency which I did not mention, but before assuming incompatibility of GR with quantum mechanics, it might make sense to doublecheck first the basic assumptions of quantum gravity (basic assumptions such as the existence of a spacetime manifold). This is why I think my thread has reason to exist.

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    Quote Originally Posted by Rinaldo View Post
    I do not understand, Strange. Are you talking about a particular theory of quantum gravity? Quantization of curved spacetime is still attempted by quite a lot of currents of quantum gravity.
    We do not know how to quantize GR. This is partly because spacetime must be continuous in GR.

    There may be a theory of quantum gravity at some point. That may or may not involve quantised spacetime. But currently all the evidence is against spacetime being quantised.

    Yes, in a more general way, it is argumented that there must not necessarily be a theory of quantum gravity. You are right that this is a restriction to my argument about inconsistency which I did not mention, but before assuming incompatibility of GR with quantum mechanics, it might make sense to doublecheck first the basic assumptions of quantum gravity (basic assumptions such as the existence of a spacetime manifold). This is why I think my thread has reason to exist.
    The spacetime manifold is the mathematical abstraction used by GR. So obviously it exists in GR.

    And, equally obviously, it might not exist in a new theory of quantum gravity. So I'm not sure what your point is.

    As you don't seem to be proposing an alternative theory I will leave others to decide if the thread should exist or not.

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    Quote Originally Posted by Strange View Post
    We do not know how to quantize GR. This is partly because spacetime must be continuous in GR.
    This mainstream statement is clearly not true, Strange, and this is the reason for the whole problem with quantum gravity. I can clearly show you in a very simple way that spacetime must not be continuous in GR:
    In 1905, Einstein developed SR, in 1908 Minkowski provided the maths for it. But explicitly, in his lecture, continuity of spacetime was a mere assumption:

    Minkowski: "In order to leave nowhere a gaping void, we imagine to ourselves that something perceptible is existent at all places and at every moment."
    The assumption of continuous spacetime was born and it is persisting up today without serious doublechecking of this assumption.
    You see that the continuity of spacetime is not part of SR.
    The same must be true for the gravitational effects of GR which are warping worldlines, but not mandatorily warping a manifold.

    But currently all the evidence is against spacetime being quantised.
    If there are doubts on quantization of spacetime, it is the moment for checking the very basic assumption of many theories of quantum gravity which is the continuity of spacetime in the form of a Lorentzian manifold.

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    Quote Originally Posted by Reality Check View Post
    …nonsense…
    Could you specify please, Reality Check? I don't see any "nonsense" in my explanations concerning the history of signatures.

    IF02: Give your derivation that Lorentzian manifolds are not compatible with GR using valid math and physics, Rinaldo.
    In the following 4 posts, you will find not less than 4 derivations, each of them proving the incompatibility of the assumption of Lorentzian manifolds with GR.

    What you have stated is that the mainstream Lorentzian manifolds are not compatible with GR even though they have been in GR for 104 years and no one has ever noticed it!
    That is exactly the claim of my thread.

    No vacuum irrelevancy - a Lorentzian manifold is a mathematical object.
    No, I contest the mathematical possibility of Lorentzian manifolds.

    No world line irrelevancy - world lines are paths in a manifold, not an entire manifold.
    Worldlines are no manifold. Threedimensional space is a manifold. Worldlines in 3D space may be parameterized by their respective proper time parameter. In a second step, observers may observe worldlines parameterized by the observer's coordinate time within the coordinates of the observer.
    (worldlines parameterized by their respective proper time parameter = worldlines before time dilation,
    worldlines parameterized by the observer's coordinate time = worldlines after time dilation)

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    A. Physical derivations: Derivation 1/4

    Note: This thread does not concern cosmology, but it refers only to gravitational effects. This is why in the following, GR is denoted by "The theory of gravity of GR".

    The SR approach: Lorentzian manifolds are not defined by the postulates of SR
    a) SR: Flat Minkowski spacetime is describing special relativity, and it is governed by only two postulates (see #1):

    - The laws of physics are the same in all inertial reference frames.
    - Speed of light is measured with the same value c in all inertial reference frames.
    Manifestly, both postulates are talking about inertial reference frames (and also about lightlike phenomena), but not about the vacuum between worldlines.
    That means that the postulates of SR are defining timelike and lightlike worldlines, but they are not defining any Lorentzian spacetime manifold. Minkowski spacetime is introducing a mathematical description of the postulates of SR, but it is only a tool for description, it is not deriving from SR even if it reflects the principles of SR at a high level. Minkowski spacetime includes Minkowski's explicit assumption of continuity of spacetime. But this assumption of Minkowski is not part of the principles of SR which are limited exclusively to the two postulates of SR.

    b) GR: The theory of gravity of GR describes the effect of gravity by curvature of spacetime. The deformation is concerning the timelike and lightlike worldlines without need for a continuous spacetime. Again, the mathematical model introduced by Einstein for the description of the theory of gravity of GR included a continuous spacetime manifold. But this is not required for the theory of gravity of GR because it is a simple deformation (stretching/ contraction) of the basic elements of SR which are - as we saw - worldlines.

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    A. Physical derivations: Derivation 2/4

    In order to know what vacuum is we must refer to quantum physics, or possibly to cosmology (with respect to dark energy), but the theory of gravity of GR provides no particular insight about the spacetime between worldlines. In particular, spacelike spacetime intervals correspond to forbidden displacements (because requiring a superluminal speed) or forbidden rigidity (cf. the current examples of Lorentz contraction with rigid rods, which, according to GR are not rigid but consisting of individual particles).

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    A. Physical derivations: Derivation 3/4

    The theory of gravity of GR does not require a Lorentzian spacetime manifold. Even if the spacetime manifold was assumed already by Minkowski in 1908, this concept is not necessary because theory of gravity of GR is a theory of the interaction of particles and fields, that means of timelike and lightlike worldlines. Even if an electromagnetic field or gravity curvature are described as continuous, their perceptible effect happens only where other worldlines are met. The only domain at my knowledge where the assumption of a continuous Lorentzian spacetime manifold is required is the domain of the theories of quantum gravity which are trying to quantize spacetime, without success.

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    B) Mathematical derivation Derivation 4/4

    For the derivation, among the 4 possible conventions, I choose the sign convention where proper time is considered to be a spacetime interval (that is +,-,-,-, with extraction of the root) and I find that only timelike intervals are real:

    I take an arbitrary Lorentzian spacetime, and I use the equation



    which provides imaginary results for spacelike spacetime intervals.
    I search for any mathematical definition of mainstream which could justify the consideration of such imaginary results, and I find none.

    The result: Spacelike spacetime intervals are not defined
    That means that in a Lorentzian spacetime, no point has any spacelike neighborhood. That means that Lorentzian spacetimes are not continuous in spacelike direction. That means that a Lorentzian spacetime is no manifold, QED.

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    Quote Originally Posted by Rinaldo View Post
    This mainstream statement is clearly not true, Strange, and this is the reason for the whole problem with quantum gravity. I can clearly show you in a very simple way that spacetime must not be continuous in GR:
    In 1905, Einstein developed SR, in 1908 Minkowski provided the maths for it. But explicitly, in his lecture, continuity of spacetime was a mere assumption:
    That doesn't say anything about it being an assumption. But also, the spacetime manifold must be continuous:

    "In differential geometry, a pseudo-Riemannian manifold,[1][2] also called a semi-Riemannian manifold, is a differentiable manifold"
    https://en.wikipedia.org/wiki/Pseudo...nnian_manifold


    If there are doubts on quantization of spacetime, it is the moment for checking the very basic assumption of many theories of quantum gravity which is the continuity of spacetime in the form of a Lorentzian manifold.
    There are theories where spacetime is discrete. But, so far, all experiments to test the idea show it to be continuous.

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    Quote Originally Posted by Rinaldo View Post
    Could you specify please, Reality Check?
    My posts have been clear - it is nonsense to compare the results from one convention to another. See my analogy of comparing a value in inches to its conversion to centimeters and expecting them to be the same. In the next posts I read ignorance, irrelevancy, argument from incredibility and opinions, not derivations.
    1 a) Lorentzian manifolds are nothing to do with the postulates of SR.
    1 b) Description of gravity in GR. A repeat of your world line = manifold ignorance.
    2. More vacuum ignorance.
    3) You have a source that states The theory of gravity of GR actually incudes a Lorentzian manifold. The reason for this is extremely simple - GR is a relativistic theory that has to be Lorentz invariant (it is based on SR!). The spacetime is used has to be Lorentz invariant. A Lorentzian manifold is Lorentz invariant.
    4) Irrelevance about conventions.

    Questions about your assertions about the mainstream Lorentzian manifold and GR.
    IF01: What happens to the Lorentzian manifold according to the definition of a Lorentzian manifold, Rinaldo?
    IF02: Give your derivation that Lorentzian manifolds are not compatible with GR using valid math and physics, Rinaldo.

    The argument from incredibility is stating the mainstream SR spacetime interval where s2 being negative is well known and even given a name ("If s2 is negative, the spacetime interval is spacelike, meaning that two events are separated by more space than time."). Being incredulous that the square root of a negative value is a complex number does not make math or physics wrong. No "justification" is needed for basic mathematics.
    Last edited by Reality Check; 2019-Dec-10 at 08:26 PM.

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    Quote Originally Posted by Reality Check View Post
    My posts have been clear - it is nonsense to compare the results from one convention to another. See my analogy of comparing a value in inches to its conversion to centimeters and expecting them to be the same.
    If I asked you to specify, Reality Check, it was because your posts were not clear. Could you specify please were I compared results from different conventions, and why in #6 you talk about " 'The history of a hidden error' nonsense"?

    1 a) Lorentzian manifolds are nothing to do with the postulates of SR.
    What do you mean? Isn't it true that Minkowski introduced a Lorentzian manifold for the description of SR?

    1 b) Description of gravity in GR. A repeat of your world line = manifold ignorance.
    You seem to reproach me that I would consider the set of worldlines to be equal to a manifold. Could you cite please where I did this?

    2. More vacuum ignorance.
    I am sorry, but the void between particles is called vacuum. Vacuum points are places to which the two postulates of SR do not refer, so the collection of points to which SR refers cannot be a manifold.

    3) GR is a relativistic theory that has to be Lorentz invariant (it is based on SR!). The spacetime is used has to be Lorentz invariant. A Lorentzian manifold is Lorentz invariant.
    Yes, Lorentz invariance is an important point, and as I claim that Lorentz manifolds do not exist, I must replace them by another Lorentz invariant solution. Such a solution exists:

    A threedimensional space manifold is Lorentz invariant. And also, particle worldlines are Lorentz invariant if each of them is parameterized by its respective proper time. That means that the Lorentzian spacetime manifold is replaced by:
    Worldlines, each of them parameterized by its respective proper time, in a space manifold.

    4) Irrelevance about conventions.
    In my mathematical derivation, I choose one of the available conventions. As there are several different sign conventions, this choice is an important step of the derivation.

    Being incredulous that the square root of a negative value is a complex number does not make math or physics wrong. No "justification" is needed for basic mathematics.
    This is the essential question. For the answer, I first would exclude any complex spacetime manifold - nobody assumes that the spacetime of GR is complex. Mainstream has implicitly chosen the other solution: splitting up spacetime into timelike and spacelike intervals, and so the essential question is if it is possible or not to provide Lorentzian metric with such a twofold interval. There are good reasons to believe that this is an error:

    a) Even if Mainstream provided the notions "timelike" and "spacelike", there has been no modification with respect to the equation of the metric



    which is a unique metric for timelike and spacelike intervals. That means: Spacelike intervals receive imaginary results. As stated above, spacetime has never been considered complex, that means that imaginary results cannot be taken into account.

    b) A mathemathical intervention (including the partitioning of spacetime in two sections) would have required a physical justification. It is a rule of nature (with possible exceptions) that nature does not use functions consisting of two different functions which are pasted together.

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    Quote Originally Posted by Strange View Post
    There are theories where spacetime is discrete. But, so far, all experiments to test the idea show it to be continuous.
    I don't think, Strange, that there has ever been an experiment to test spacelike continuity of spacetime. I claim: worldlines are continuous, but there is no continuity between them in spacelike direction.

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    Quote Originally Posted by Rinaldo View Post
    I don't think, Strange, that there has ever been an experiment to test spacelike continuity of spacetime.
    I don't know what that means. But there have certainly been experiments to test if space is discrete or continuous. So far all the evidence is with continuous.

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    Quote Originally Posted by Rinaldo View Post
    I am sorry, but the void between particles is called vacuum. Vacuum points are places to which the two postulates of SR do not refer, so the collection of points to which SR refers cannot be a manifold.
    This is incoherent nonsense. You can, I suppose, choose to define vacuum as meaning the space between particles but it is a rather odd, non-standard definition. But that is OK because it is irrelevant.

    What are "vacuum points"? And why do you think the postulates of SR do not apply to them?

    SR applies just as much in a vacuum as it does in the presence of matter. The coordinate system does not depend on the presence (or absence) of matter.

    It is a rule of nature (with possible exceptions) that nature does not use functions consisting of two different functions which are pasted together.
    Is it? Where is this rule written? (I have no idea what "two different functions which are pasted together" means, either.)

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    Quote Originally Posted by Strange View Post
    I don't know what that means. But there have certainly been experiments to test if space is discrete or continuous. So far all the evidence is with continuous.
    Spacelike is the opposite of timelike, Strange. Spacelike movements would require speed beyond speed of light. Spacelike intervals are imaginary. My mathematical derivation in # 15 shows that spacelike intervals are not defined, and that Lorentzian manifolds are not continuous in spacelike direction. I don't think that there is experimental evidence for continuity of spacetime in spacelike direction.

    In contrast, the threedimensional space manifold is continuous.

    Example: A 20 cm ruler at time t0: The space interval is 20 cm, the spacetime interval is spacelike, it is 20i cm, that means that the spacetime interval is imaginary, it is not defined. The space interval provides continuity, the imaginary spacetime interval does not.

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    Quote Originally Posted by Strange View Post
    This is incoherent nonsense. You can, I suppose, choose to define vacuum as meaning the space between particles but it is a rather odd, non-standard definition.
    Itís the definition of Wikipedia, Strange: "Vacuum is space devoid of matter."

    What are "vacuum points"? And why do you think the postulates of SR do not apply to them?
    The two postulates are:
    - The laws of physics are the same in all inertial reference frames.
    - Speed of light is measured with the same value c in all inertial reference frames.

    Manifestly, both postulates are talking about inertial reference frames (and also about lightlike phenomena), but not about the vacuum between worldlines.

    In short, the two postulates "don't care" of any point which is not found on a worldline, or in mathematical language: vacuum is not defined by the two postulates of SR.

    Where is this rule written? (I have no idea what "two different functions which are pasted together" means, either.)
    It was simply an allusion to quantum mechanics with its complex functions. Concretely I meant: The spacetime intervals of a Lorentzian manifold are always imaginary in one direction, either in timelike direction or in spacelike direction. Thus, if we want to get a Lorentzian manifold whose timelike and spacelike intervals are real, we would have to take two Lorentzian manifolds, one with real values in timelike direction and the other with real values in spacelike direction, cut out the real half of each of them and paste together these both halfs of two different Lorentzian manifolds. That is not natural and requires justification.

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    Quote Originally Posted by Rinaldo View Post
    Spacelike intervals are imaginary.
    Well, thanks for confirming that you don't have a clue what you are talking about.

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    Quote Originally Posted by Rinaldo View Post
    It’s the definition of Wikipedia, Strange: "Vacuum is space devoid of matter."
    That has nothing to do with what you said. And says nothing about SR not applying in a vacuum.

    The two postulates are:
    - The laws of physics are the same in all inertial reference frames.
    - Speed of light is measured with the same value c in all inertial reference frames.

    Manifestly, both postulates are talking about inertial reference frames (and also about lightlike phenomena), but not about the vacuum between worldlines.
    It doesn't mention vacuum because it is completely irrelevant.

    What does "vacuum between worldlines" mean? You are just posting random sentences now.

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    Quote Originally Posted by Strange View Post
    It doesn't mention vacuum because it is completely irrelevant.
    Exactly, Strange. That is the point. The postulates do not pronounce any prohibition to be applied to vacuum between worldlines, but they simply don't care about vacuum between worldlines. They concern only worldlines of particles and lightlike phenomena.
    Between worldlines of particles there is void. That is what I call "vacuum between worldlines".

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    Quote Originally Posted by Rinaldo View Post
    Exactly, Strange. That is the point. The postulates do not pronounce any prohibition to be applied to vacuum between worldlines, but they simply don't care about vacuum between worldlines. They concern only worldlines of particles and lightlike phenomena.
    Between worldlines of particles there is void. That is what I call "vacuum between worldlines".
    I can't give a scientific or logical answer to this. It is complete and utter gibberish. It is solidly in the realm of "not even wrong".

    There is no such thing as "vacuum between worldlines." So you are claiming that SR does not apply to a thing that does not exist.

    Pretty much everything else you have written is equally meaningless. You clearly do not understand SR or even the simple mathematical concepts behind it.

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    Quote Originally Posted by Rinaldo View Post
    ...
    It is not true that Minkowski introduced a Lorentzian manifold from the postulates of SR which is your assertion. He noted that space + time = space rime and that it has to be Lorenz invariant thus what we call a Minkowski spacetime.
    Your "vacuum" ignorance again. It is persisting with world lines along with "vacuum" that suggests you think that these are the entire manifolds. You are making obvious errors about manifolds. Think about Euclidean space in math and used in classical physics. This is a continuous space with an infinite number of points between each point. No vacuum there! Manifolds are a generalization of Euclidean space. They are mathematical. They are defined to be continuous. They do not have vacuum.
    IF03: Give your sources that say that a Lorentzian manifold must have "vacuum" or "void(s)", Rinaldo.

    IF04: State where the 2 postures of SR exclude ("do not apply to") "vacuum points" (presumably points in physical vacuums), Rinaldo.
    Special relativity
    it is based on two postulates:
    1.the laws of physics are invariant (i.e. identical) in all inertial frames of reference (i.e. non-accelerating frames of reference); and
    2.the speed of light in a vacuum is the same for all observers, regardless of the motion of the light source or observer
    You persist with a blatant error that a Lorentzian manifold is not compatible with GR when it is easy see that it is from reading about GR. Differentiable manifolds in physics have to be continuous because we want to apply calculus and calculus needs a continuous space. A simply put example: a speed dx/dt is a limit as finite intervals in x and t tend to 0. Those intervals cannot come to a screeching halt at a finite value! GR has the additional requirement that its manifold be Lorentz invariant.

    IF05a: Show that "A threedimensional space manifold is Lorentz invariant" for example the Euclidean space used in classical physics, Rinaldo.
    IF05b: Show that "particle worldlines are Lorentz invariant if each of them is parameterized by its respective proper time", Rinaldo.
    IF05c: Give your definition of calculus in your "world line only manifold" and show that it works, Rinaldo.

    Quote Originally Posted by Rinaldo View Post
    This is the essential question.
    There is no "essential question". There is the mathematical fact that a spacetime interval equation exists and sometimes gives a negative number so that s = a complex number. You were incredulous or surprised about that. So I wrote: "Being incredulous that the square root of a negative value is a complex number does not make math or physics wrong. No "justification" is needed for basic mathematics.".
    Spacelike spacetime intervals are defined as being negative. The square root of a spacetime interval (spacetime distance?) is then a complex number. No one says that gives a complex spacetime because they can look at the definition of spacetime and see it has no complex numbers!

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    Quote Originally Posted by Rinaldo View Post
    Example: A 20 cm ruler at time t0: The space interval is 20 cm, the spacetime interval is spacelike, it is 20i cm, that means that the spacetime interval is imaginary, it is not defined. The space interval provides continuity, the imaginary spacetime interval does not.
    This example neatly sums up what you don't seem to understand about a spacetime interval. Spacetime intervals are defined between events. All your equation is actually saying here is that if you have two events occur that are spatially separated but not temporally then their interval is space-like.

    Also spacetime intervals are not usually square rooted. S^2 is the interval, not s.

    Pretty much everything you have said about spacetime intervals is wrong because of these points.

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    Quote Originally Posted by Shaula View Post
    Also spacetime intervals are not usually square rooted. S^2 is the interval, not s.
    There are both conventions, Shaula. Some authors extract the root of the square and some don't.
    The latter are fearing the imaginary result, but the physical issue of spacelike spacetime intervals persists.

    All your equation is actually saying here is that if you have two events occur that are spatially separated but not temporally then their interval is space-like.
    In particular it says that spacelike spacetime intervals are imaginary (or negative squares).

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