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  1. #1
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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.

  2. #2
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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.

  5. #5
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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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    Rinaldo, it is about time that you start presenting something real. We have now two pages of discussion whether s or s2 should be used by mainstream, but there is NOTHING that you write about the rest YOUR theory. I see no real purpose in keeping this thread open. Please, start discussing someting significant, including the math, of your theory in your following posts. I think we have chewed enough on this s-s2 topic.
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    Quote Originally Posted by tusenfem View Post

    Rinaldo, it is about time that you start presenting something real. We have now two pages of discussion whether s or s2 should be used by mainstream, but there is NOTHING that you write about the rest YOUR theory. I see no real purpose in keeping this thread open. Please, start discussing someting significant, including the math, of your theory in your following posts. I think we have chewed enough on this s-s2 topic.
    Thank you for your feedback, Tusenfem.
    Concerning the s-s2 topic, I think that all has been said, but personally, it is not my impression that two pages have been wasted for the s-s2 conventions. All the contrary, I would like to thank here the questioners for their critical questions on a high level which are helping me enormously with the clarification of my theory. It is true that we have talked much about conventions in the mathematical part, and perhaps we can now talk about mathematical arguments which could be opposed to my theory.

    Here is a new start of the thread, I hope that it is interesting enough for this forum.

    At stake: This thread is about why theories of quantum gravity are not working. It is a quite important topic because much scientific manpower is currently bound by the attempts to quantize spacetime on the one hand and by the research for new spacetime structures for the modification of GR on the other hand. - According to my theory, all this is unnecessary because GR is harmonizing very well with quantum mechanics, it is only our concept of Lorentzian spacetime which is not entirely complying with GR.

    It took me several years to come to a result because at first sight, everything seems perfectly without flaw, and the maths of GR are perfectly corresponding to experimental evidence. In this thread I show that nevertheless there is an incoherence of the maths of GR at a very precise point - that is my theory. The incoherence is hard to see because of the different conventions we discussed in extension, but once all conventions have been put aside it is hard to contest.

    Mathematically, my derivation is very short and clear, as I have shown (see #15): Lorentzian spacetime is real, but spacelike spacetime intervals are imaginary, that means that they do not exist. My theory says that this is a clear mathematical flaw. This is the whole maths.

    Wikipedia confirms indirectly that there is a problem, by saying that negative square roots are avoided by different conventions. And accordingly, the questioners put forward conventions, but I think that conventions cannot be used as the only arguments against my mathematical derivation. Now my theory should be appreciated with respect to mathematical arguments.

    Physical arguments are corroborating the mathematical result, as shown in ##12 to 14, in particular, vacuum between worldlines is not defined by GR but by quantum physics.

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    this is nothing different and will lead to the exact same discussion about s-s2 and how to interpret imaginary (mathematical meaning, not general language meaning as "non existent/only in your mind") results.

    closed for moderator discussion
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    Okay, after a bit of discussion, I think this is the best way to go here.
    We start off from scratch and now ONLY the "real manifold" is going to be discussed, and looked at how this works for GR.
    There will be no more discussions about the usual manifold, and the questions whether there are real or imaginary distances and all that stuff.
    rinaldo will show how GR works with only real numbers.
    Happy discussion!
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    My thread is no theory in the sense of an alternative idea, it concerns the weak point of many current theories of quantum gravity.

    My thread seems to concern the only domain of physics which seems to be governed by a bunch of different conventions and concepts without anybody worrying about possible consequences. I cannot explain why there is such a lack of interest among physicists, and in the same way I don't know why there is such a lack of interest in this forum.

    Theories of quantum gravity are becoming more and more ingenious, but without success, and it is my prediction that there will be no theory of quantum gravity possible without clearing up the sign conventions and abandoning the concept of continuous Lorentzian manifolds.

    If there is no further interest in discussion, I hereby ask to close this thread.

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    Quote Originally Posted by Rinaldo View Post
    My thread seems to concern the only domain of physics which seems to be governed by a bunch of different conventions and concepts without anybody worrying about possible consequences. I cannot explain why there is such a lack of interest among physicists, and in the same way I don't know why there is such a lack of interest in this forum.
    The trouble is that your thread is based on a series of misunderstandings of current theory.

    There is a great deal of interest among physicists (and probably on this forum) in theories of quantum gravity. Some of which require a replacement of GR with something where spacetime is not continuous. Or, in some cases, doesn't exist at all as the background of the theory, but only emerges as a consequence of the theory.

    So it appears you are making a strawman argument by insisting that you are the only person who is think that GR, with its continuous spacetime, may need to be replaced. In fact many people, with the requisite expertise, are thinking about this problem.

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    The "lack of interest", Strange, is referring to the diverging sign conventions and the problems which are hidden by them (imaginary spacelike spacetime intervals), this is precisely what we talked about in this thread, physicists seem not to be interested to clear up this topic, and I learned with regret that also in this forum, this is not considered any longer to be an interesting topic.

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    Quote Originally Posted by Rinaldo View Post
    My thread seems to concern the only domain of physics which seems to be governed by a bunch of different conventions and concepts without anybody worrying about possible consequences. I cannot explain why there is such a lack of interest among physicists, and in the same way I don't know why there is such a lack of interest in this forum.
    As Strange says - it is because most of your arguments are based either on misconceptions or unsupported assertions.

    If GR and SR are so deficient as you claim then this should be an easy challenge:
    Please show, in detail and in full, an SR or GR calculation using a Lorentz manifold that leads to a direct contradiction with observations that is due to one of your claims about the deficiencies of a Lorentz manifold. Not just with words, not just with arguments from incredulity - show us a calculation you have made that directly exposes this weakness by contradicting observations.
    Please show, in detail and in full, how your assertion that worldlines are the only valid thing in GR or SR fixes this inconsistency.

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    Quote Originally Posted by Shaula View Post
    As Strange says - it is because most of your arguments are based either on misconceptions or unsupported assertions.

    If GR and SR are so deficient as you claim then this should be an easy challenge:
    Please show, in detail and in full, an SR or GR calculation using a Lorentz manifold that leads to a direct contradiction with observations that is due to one of your claims about the deficiencies of a Lorentz manifold. Not just with words, not just with arguments from incredulity - show us a calculation you have made that directly exposes this weakness by contradicting observations.
    Please show, in detail and in full, how your assertion that worldlines are the only valid thing in GR or SR fixes this inconsistency.

    Okay, let me make that official. rinaldo, it is time to put your money where your mouth is, and this is a good start. Please proceed with an enlightening reply, including the math.
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    Quote Originally Posted by tusenfem View Post

    Okay, let me make that official. rinaldo, it is time to put your money where your mouth is, and this is a good start. Please proceed with an enlightening reply, including the math.
    I tried my best, Tusenfem.

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    Quote Originally Posted by Shaula View Post
    Please show, in detail and in full, an SR or GR calculation using a Lorentz manifold that leads to a direct contradiction with observations that is due to one of your claims about the deficiencies of a Lorentz manifold. Not just with words, not just with arguments from incredulity
    It is curved spacetime which is in direct contradiction with observations (of quantum mechanics), Shaula. This is the result of attempts to quantize curved spacetime, it is the current state of mainstream

    The mathematical calculation leading to the direct contradiction is shown in my #15 - imaginary results in real spacetime are invalid - no additional calculation is required (if you do not agree please tell me what is missing, but an invalid result is an invalid result).

    show us a calculation you have made that directly exposes this weakness by contradicting observations.

    This is shown by the theories of quantum gravity which are trying to foliate spacetime into spacelike hypersurfaces, they are failing, and sometimes they claim that GR must be modified, in spite of the experimental confirmation of GR.

    Please refer to these attempts of the theories of quantum gravity, I am not able to make similar sophisticated calculations which are leading nowhere.

    Please show, in detail and in full, how your assertion that worldlines are the only valid thing in GR or SR fixes this inconsistency.
    Worldlines of particles and fields are timelike and lightlike. The theory of gravity of GR refers only to worldlines (vacuum is not defined by it), that means that it is not based on spacelike intervals.

    It seems that, at the exception of the theories of quantum gravity, the whole relevant physics of the theory of gravity of GR concern only timelike and lightlike worldlines such that there is no issue.
    Last edited by Rinaldo; 2019-Dec-23 at 09:10 PM.

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    Quote Originally Posted by Rinaldo View Post
    It is curved spacetime which is in direct contradiction with observations (of quantum mechanics), Shaula. This is the result of attempts to quantize curved spacetime, it is the current state of mainstream
    Which observations? Please give details. Of observations that QM and GR predict and get wrong, not the issues with unifying the two.

    Curved spacetime is just fine QM: https://en.wikipedia.org/wiki/Quantu...rved_spacetime - it is the quantisation of gravity and a desire to quantise spacetime that lead to issues, not curved spacetime.

    Quote Originally Posted by Rinaldo View Post
    The mathematical calculation leading to the direct contradiction is shown in my #15 - imaginary results in real spacetime are invalid - no additional calculation is required (if you do not agree please tell me what is missing, but an invalid result is an invalid result).
    Please show, in detail and in full, an SR or GR calculation using a Lorentz manifold that leads to a direct contradiction with observations that is due to one of your claims about the deficiencies of a Lorentz manifold. Not just with words, not just with arguments from incredulity - show us a calculation you have made that directly exposes this weakness by contradicting observations.

    Your argument that you don't think a negative spacetime interval is valid is an assertion you have made with no evidence. Which is why I have asked you to provide a calculation that leads to an observation that is at odds with what we have seen in experiments. "I don't like negative spacetime intervals because if you take a square root they go imaginary and I don't like that" is a misconception about what the spacetime interval is defined as combined with an assertion about what is valid based on an argument from incredulity. In other words your argument is a belief you have and cannot provide evidence for. Think I'm wrong? Show me, like I asked you to.

    Quote Originally Posted by Rinaldo View Post
    This is shown by the theories of quantum gravity which are trying to foliate spacetime into spacelike hypersurfaces, they are failing, and sometimes they claim that GR must be modified, in spite of the experimental confirmation of GR.

    Please refer to these attempts of the theories of quantum gravity, I am not able to make similar sophisticated calculations which are leading nowhere.
    So you understand these theories well enough to define what their issues are but not well enough to show us any details about this. Sorry, that's unconvincing. Sounds like all you have are just so stories.

    Quote Originally Posted by Rinaldo View Post
    Worldlines of particles and fields are timelike and lightlike. The theory of gravity of GR refers only to worldlines (vacuum is not defined by it), that means that it is not based on spacelike intervals.

    It seems that, at the exception of the theories of quantum gravity, the whole relevant physics of the theory of gravity of GR concern only timelike and lightlike worldlines such that there is no issue.
    More empty assertions and patently false claims. GR and SR deal with spacelike spacetime intervals without any issues. Unless you can provide an example calculation that leads to a prediction that doesn't match observations then this is more of your baseless claims.

    You've not even come close answering my questions. All you have presented is a melange of misconceptions, assertions and opinions. This is not how physics works.

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