Can matter be described as the result of the curvature of space, instead of vice versa?












14














Can matter be described as the result of the curvature of space, rather than the curvature of space being the result of matter, and energy being the cause of the curvature of space?










share|cite|improve this question




















  • 1




    One difficulty that has not yet been addressed in the answers is that matter has to obey quantum mechanics, but it is not yet clear how to get the curvature of spacetime to obey quantum mechanics.
    – Display Name
    Dec 5 at 14:59










  • speculation: Maybe describing something as "matter" or "spacetime curvature" is ultimately a matter of taste, like say a particle or wave representation.
    – R. Rankin
    Dec 6 at 10:01






  • 1




    One historical tidbit: Einstein & Rosen (1935) represented an electron as Schwarzschild's curved spacetime. However in retrospect, we reinterpret their work as a "wormhole" solution with nothing to do with an electron.
    – Colin MacLaurin
    Dec 12 at 0:18
















14














Can matter be described as the result of the curvature of space, rather than the curvature of space being the result of matter, and energy being the cause of the curvature of space?










share|cite|improve this question




















  • 1




    One difficulty that has not yet been addressed in the answers is that matter has to obey quantum mechanics, but it is not yet clear how to get the curvature of spacetime to obey quantum mechanics.
    – Display Name
    Dec 5 at 14:59










  • speculation: Maybe describing something as "matter" or "spacetime curvature" is ultimately a matter of taste, like say a particle or wave representation.
    – R. Rankin
    Dec 6 at 10:01






  • 1




    One historical tidbit: Einstein & Rosen (1935) represented an electron as Schwarzschild's curved spacetime. However in retrospect, we reinterpret their work as a "wormhole" solution with nothing to do with an electron.
    – Colin MacLaurin
    Dec 12 at 0:18














14












14








14


9





Can matter be described as the result of the curvature of space, rather than the curvature of space being the result of matter, and energy being the cause of the curvature of space?










share|cite|improve this question















Can matter be described as the result of the curvature of space, rather than the curvature of space being the result of matter, and energy being the cause of the curvature of space?







general-relativity energy spacetime curvature matter






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 5 at 14:19









knzhou

41.7k11117199




41.7k11117199










asked Dec 5 at 10:13









Kane

713




713








  • 1




    One difficulty that has not yet been addressed in the answers is that matter has to obey quantum mechanics, but it is not yet clear how to get the curvature of spacetime to obey quantum mechanics.
    – Display Name
    Dec 5 at 14:59










  • speculation: Maybe describing something as "matter" or "spacetime curvature" is ultimately a matter of taste, like say a particle or wave representation.
    – R. Rankin
    Dec 6 at 10:01






  • 1




    One historical tidbit: Einstein & Rosen (1935) represented an electron as Schwarzschild's curved spacetime. However in retrospect, we reinterpret their work as a "wormhole" solution with nothing to do with an electron.
    – Colin MacLaurin
    Dec 12 at 0:18














  • 1




    One difficulty that has not yet been addressed in the answers is that matter has to obey quantum mechanics, but it is not yet clear how to get the curvature of spacetime to obey quantum mechanics.
    – Display Name
    Dec 5 at 14:59










  • speculation: Maybe describing something as "matter" or "spacetime curvature" is ultimately a matter of taste, like say a particle or wave representation.
    – R. Rankin
    Dec 6 at 10:01






  • 1




    One historical tidbit: Einstein & Rosen (1935) represented an electron as Schwarzschild's curved spacetime. However in retrospect, we reinterpret their work as a "wormhole" solution with nothing to do with an electron.
    – Colin MacLaurin
    Dec 12 at 0:18








1




1




One difficulty that has not yet been addressed in the answers is that matter has to obey quantum mechanics, but it is not yet clear how to get the curvature of spacetime to obey quantum mechanics.
– Display Name
Dec 5 at 14:59




One difficulty that has not yet been addressed in the answers is that matter has to obey quantum mechanics, but it is not yet clear how to get the curvature of spacetime to obey quantum mechanics.
– Display Name
Dec 5 at 14:59












speculation: Maybe describing something as "matter" or "spacetime curvature" is ultimately a matter of taste, like say a particle or wave representation.
– R. Rankin
Dec 6 at 10:01




speculation: Maybe describing something as "matter" or "spacetime curvature" is ultimately a matter of taste, like say a particle or wave representation.
– R. Rankin
Dec 6 at 10:01




1




1




One historical tidbit: Einstein & Rosen (1935) represented an electron as Schwarzschild's curved spacetime. However in retrospect, we reinterpret their work as a "wormhole" solution with nothing to do with an electron.
– Colin MacLaurin
Dec 12 at 0:18




One historical tidbit: Einstein & Rosen (1935) represented an electron as Schwarzschild's curved spacetime. However in retrospect, we reinterpret their work as a "wormhole" solution with nothing to do with an electron.
– Colin MacLaurin
Dec 12 at 0:18










2 Answers
2






active

oldest

votes


















13














Maybe one day.



This idea, at least it's mathematical genesis seems to have begun with Riemann and later Clifford. In 1870 Clifford (a very good mathematician), building upon Riemann gave a lecture stating:




1) That small portions of space are in fact analogous to little hills
on a surface which is on the average flat namely that the ordinary
laws of geometry are not valid.



2) That this property of being curved or distorted is continually being
passed on from one portion of space to another after the matter of a
wave.



3) That this variation of curvature of space is what really happens in
that phenomena we call the motion of matter, whether ponderable or
etherial.



4)That in the physical world nothing else takes place but this
variation, subject (possibly) to the law of continuity.




This was the type of thinking that Led Einstein to consider space and time as dynamic entities and develop General Relativity(using Riemann's then developed geometry). Many others have sought to describe more of the universe than gravity through this type of program.



As of now, it's a no; however some progress has been made.



Because Electromagnetism curves spacetime like matter does, it has been shown (Rainich, Misner, Wheeler) that just the "footprints" left on spacetime by electromagnetic fields are enough to reconstruct the fields themselves (up do a "duality" rotation). This of course only holds for classical electrodynamics. This was called "Geometrodynamics" by Wheeler (who was famous for coining other phrases like blackhole and wormhole as well).



Wheeler showed analytically that a properly constructed ball of gravitational and electromagnetic radiations would possess the properties of a massive object (he called this a geon) and he applied similar topological ideas such that electric charges can appear to exist when there is in fact none (a trick of spacetime geometry).



To go further, one would need to be able to describe the gauge fields of the standard model in terms of geometry and properly quantize it. Whether these things can be found withing the topology of general relativity's spacetime, well that jury is still out.



The equations involved here are highly nonlinear, especially when you have the feedback effect of describing say the electromagnetic field through geometry which in turn effects the geometry again.



Anyway, when I read your question I am literally reading a book in front of my face:



"The Geometrodynamics of Gauge Fields. On the geometry of Yang-Mills
fields and Gravitational gauge theories"
Eckehard W. Mielke



I'll update this when I'm done maybe, it's an excellent book by the way.



PS: I had the pleasure of seeing Kip Thorne Lecture this (last?) year and he seems a large proponent of geometrodynamics. As one of the founders of LIGO (we're FINALLY detecting gravitational waves!!! still can't believe it) there's a potential to test such theories in the foreseeable future.






share|cite|improve this answer































    5














    The answer is yes !



    In General Relativity, there's a not very well known theorem called the Campbell-Magaard theorem that states that any metric in 4D spacetime (including matter) could be represented (or embeded) as a metric in empty 5D spacetime (Ricci flat spacetime, wich implies pure geometry) :



    https://en.wikipedia.org/wiki/Campbell%27s_theorem_(geometry)



    https://arxiv.org/abs/gr-qc/0302015



    There are several simple solutions to Einstein's equation in 5D vacuum ($R_{AB}^{(5)} = 0$) that represent matter in 4D ($R_{mu nu}^{(4)} ne 0$). This could be interpreted as matter made of pure geometry, in higher dimensions spacetimes.



    Here's a very simple example. Consider the following metric in 5D spacetime ($theta$ is a cyclic coordinate in the fifth dimension) :
    begin{equation}tag{1}
    ds_{(5)}^2 = dt^2 - a^2(t) (dx^2 + dy^2 + dz^2) - b^2(t) , dtheta^2.
    end{equation}

    Substitute this metric into the 5D Einstein's equation without any matter :
    begin{equation}tag{2}
    R_{AB}^{(5)} = 0.
    end{equation}

    Then you get as a non-trivial solution these two scale factors :
    begin{align}tag{3}
    a(t) &= alpha , t^{1/2}, & b(t) &= beta , t^{- 1/2}.
    end{align}

    This is the same as pure radiation in an homogeneous 4D spacetime :
    begin{equation}tag{4}
    ds_{(4)}^2 = dt^2 - a^2(t) (dx^2 + dy^2 + dz^2).
    end{equation}

    With
    begin{equation}tag{5}
    R_{mu nu}^{(4)} - frac{1}{2} , g_{mu nu}^{(4)} , R^{(4)} = -, kappa , T_{mu nu}^{(4)},
    end{equation}

    and $T_{mu nu}^{(4)}$ describing a perfect fluid of incoherent radiation ($p_{rad} = frac{1}{3} , rho_{rad}$):



    begin{equation}tag{6}
    T_{mu nu}^{(4)} = (rho_{rad} + p_{rad}) , u_{mu}^{(4)} , u_{nu}^{(4)} - g_{mu nu}^{(4)} , p_{rad}.
    end{equation}



    This is the subject of the induced matter hypothesis, and is extremely fascinating! For more on the Campbell-Magaard theorem and the induced-matter theory :



    https://arxiv.org/abs/gr-qc/0507107



    https://arxiv.org/abs/gr-qc/0305066



    https://arxiv.org/abs/gr-qc/9907040






    share|cite|improve this answer























    • How are $rho_{rad}$ and $p_{rad}$ related to the geometric parameters $a(t)$ and $b(t)$?
      – lurscher
      Dec 5 at 14:25










    • @lurscher : $rho_{rad} propto a(t)^{- 4}$, which is the standard relation between energy density of radiation and the scale factor, in FLRW cosmologies.
      – Cham
      Dec 5 at 14:40










    • @Cham Is this a special 4d case of the Nash embedding theorem?
      – R. Rankin
      Dec 6 at 4:15










    • @R.Rankin, I don't know the Nash embedding theorem. Have you some references?
      – Cham
      Dec 6 at 12:12










    • Appears, Nash embedding is a case of whitney embedding. Here's a great stackexchange answer, but there are papers on them everywhere if your looking: math.stackexchange.com/questions/236285/…
      – R. Rankin
      Dec 6 at 12:30













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "151"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: false,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: null,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fphysics.stackexchange.com%2fquestions%2f445264%2fcan-matter-be-described-as-the-result-of-the-curvature-of-space-instead-of-vice%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    13














    Maybe one day.



    This idea, at least it's mathematical genesis seems to have begun with Riemann and later Clifford. In 1870 Clifford (a very good mathematician), building upon Riemann gave a lecture stating:




    1) That small portions of space are in fact analogous to little hills
    on a surface which is on the average flat namely that the ordinary
    laws of geometry are not valid.



    2) That this property of being curved or distorted is continually being
    passed on from one portion of space to another after the matter of a
    wave.



    3) That this variation of curvature of space is what really happens in
    that phenomena we call the motion of matter, whether ponderable or
    etherial.



    4)That in the physical world nothing else takes place but this
    variation, subject (possibly) to the law of continuity.




    This was the type of thinking that Led Einstein to consider space and time as dynamic entities and develop General Relativity(using Riemann's then developed geometry). Many others have sought to describe more of the universe than gravity through this type of program.



    As of now, it's a no; however some progress has been made.



    Because Electromagnetism curves spacetime like matter does, it has been shown (Rainich, Misner, Wheeler) that just the "footprints" left on spacetime by electromagnetic fields are enough to reconstruct the fields themselves (up do a "duality" rotation). This of course only holds for classical electrodynamics. This was called "Geometrodynamics" by Wheeler (who was famous for coining other phrases like blackhole and wormhole as well).



    Wheeler showed analytically that a properly constructed ball of gravitational and electromagnetic radiations would possess the properties of a massive object (he called this a geon) and he applied similar topological ideas such that electric charges can appear to exist when there is in fact none (a trick of spacetime geometry).



    To go further, one would need to be able to describe the gauge fields of the standard model in terms of geometry and properly quantize it. Whether these things can be found withing the topology of general relativity's spacetime, well that jury is still out.



    The equations involved here are highly nonlinear, especially when you have the feedback effect of describing say the electromagnetic field through geometry which in turn effects the geometry again.



    Anyway, when I read your question I am literally reading a book in front of my face:



    "The Geometrodynamics of Gauge Fields. On the geometry of Yang-Mills
    fields and Gravitational gauge theories"
    Eckehard W. Mielke



    I'll update this when I'm done maybe, it's an excellent book by the way.



    PS: I had the pleasure of seeing Kip Thorne Lecture this (last?) year and he seems a large proponent of geometrodynamics. As one of the founders of LIGO (we're FINALLY detecting gravitational waves!!! still can't believe it) there's a potential to test such theories in the foreseeable future.






    share|cite|improve this answer




























      13














      Maybe one day.



      This idea, at least it's mathematical genesis seems to have begun with Riemann and later Clifford. In 1870 Clifford (a very good mathematician), building upon Riemann gave a lecture stating:




      1) That small portions of space are in fact analogous to little hills
      on a surface which is on the average flat namely that the ordinary
      laws of geometry are not valid.



      2) That this property of being curved or distorted is continually being
      passed on from one portion of space to another after the matter of a
      wave.



      3) That this variation of curvature of space is what really happens in
      that phenomena we call the motion of matter, whether ponderable or
      etherial.



      4)That in the physical world nothing else takes place but this
      variation, subject (possibly) to the law of continuity.




      This was the type of thinking that Led Einstein to consider space and time as dynamic entities and develop General Relativity(using Riemann's then developed geometry). Many others have sought to describe more of the universe than gravity through this type of program.



      As of now, it's a no; however some progress has been made.



      Because Electromagnetism curves spacetime like matter does, it has been shown (Rainich, Misner, Wheeler) that just the "footprints" left on spacetime by electromagnetic fields are enough to reconstruct the fields themselves (up do a "duality" rotation). This of course only holds for classical electrodynamics. This was called "Geometrodynamics" by Wheeler (who was famous for coining other phrases like blackhole and wormhole as well).



      Wheeler showed analytically that a properly constructed ball of gravitational and electromagnetic radiations would possess the properties of a massive object (he called this a geon) and he applied similar topological ideas such that electric charges can appear to exist when there is in fact none (a trick of spacetime geometry).



      To go further, one would need to be able to describe the gauge fields of the standard model in terms of geometry and properly quantize it. Whether these things can be found withing the topology of general relativity's spacetime, well that jury is still out.



      The equations involved here are highly nonlinear, especially when you have the feedback effect of describing say the electromagnetic field through geometry which in turn effects the geometry again.



      Anyway, when I read your question I am literally reading a book in front of my face:



      "The Geometrodynamics of Gauge Fields. On the geometry of Yang-Mills
      fields and Gravitational gauge theories"
      Eckehard W. Mielke



      I'll update this when I'm done maybe, it's an excellent book by the way.



      PS: I had the pleasure of seeing Kip Thorne Lecture this (last?) year and he seems a large proponent of geometrodynamics. As one of the founders of LIGO (we're FINALLY detecting gravitational waves!!! still can't believe it) there's a potential to test such theories in the foreseeable future.






      share|cite|improve this answer


























        13












        13








        13






        Maybe one day.



        This idea, at least it's mathematical genesis seems to have begun with Riemann and later Clifford. In 1870 Clifford (a very good mathematician), building upon Riemann gave a lecture stating:




        1) That small portions of space are in fact analogous to little hills
        on a surface which is on the average flat namely that the ordinary
        laws of geometry are not valid.



        2) That this property of being curved or distorted is continually being
        passed on from one portion of space to another after the matter of a
        wave.



        3) That this variation of curvature of space is what really happens in
        that phenomena we call the motion of matter, whether ponderable or
        etherial.



        4)That in the physical world nothing else takes place but this
        variation, subject (possibly) to the law of continuity.




        This was the type of thinking that Led Einstein to consider space and time as dynamic entities and develop General Relativity(using Riemann's then developed geometry). Many others have sought to describe more of the universe than gravity through this type of program.



        As of now, it's a no; however some progress has been made.



        Because Electromagnetism curves spacetime like matter does, it has been shown (Rainich, Misner, Wheeler) that just the "footprints" left on spacetime by electromagnetic fields are enough to reconstruct the fields themselves (up do a "duality" rotation). This of course only holds for classical electrodynamics. This was called "Geometrodynamics" by Wheeler (who was famous for coining other phrases like blackhole and wormhole as well).



        Wheeler showed analytically that a properly constructed ball of gravitational and electromagnetic radiations would possess the properties of a massive object (he called this a geon) and he applied similar topological ideas such that electric charges can appear to exist when there is in fact none (a trick of spacetime geometry).



        To go further, one would need to be able to describe the gauge fields of the standard model in terms of geometry and properly quantize it. Whether these things can be found withing the topology of general relativity's spacetime, well that jury is still out.



        The equations involved here are highly nonlinear, especially when you have the feedback effect of describing say the electromagnetic field through geometry which in turn effects the geometry again.



        Anyway, when I read your question I am literally reading a book in front of my face:



        "The Geometrodynamics of Gauge Fields. On the geometry of Yang-Mills
        fields and Gravitational gauge theories"
        Eckehard W. Mielke



        I'll update this when I'm done maybe, it's an excellent book by the way.



        PS: I had the pleasure of seeing Kip Thorne Lecture this (last?) year and he seems a large proponent of geometrodynamics. As one of the founders of LIGO (we're FINALLY detecting gravitational waves!!! still can't believe it) there's a potential to test such theories in the foreseeable future.






        share|cite|improve this answer














        Maybe one day.



        This idea, at least it's mathematical genesis seems to have begun with Riemann and later Clifford. In 1870 Clifford (a very good mathematician), building upon Riemann gave a lecture stating:




        1) That small portions of space are in fact analogous to little hills
        on a surface which is on the average flat namely that the ordinary
        laws of geometry are not valid.



        2) That this property of being curved or distorted is continually being
        passed on from one portion of space to another after the matter of a
        wave.



        3) That this variation of curvature of space is what really happens in
        that phenomena we call the motion of matter, whether ponderable or
        etherial.



        4)That in the physical world nothing else takes place but this
        variation, subject (possibly) to the law of continuity.




        This was the type of thinking that Led Einstein to consider space and time as dynamic entities and develop General Relativity(using Riemann's then developed geometry). Many others have sought to describe more of the universe than gravity through this type of program.



        As of now, it's a no; however some progress has been made.



        Because Electromagnetism curves spacetime like matter does, it has been shown (Rainich, Misner, Wheeler) that just the "footprints" left on spacetime by electromagnetic fields are enough to reconstruct the fields themselves (up do a "duality" rotation). This of course only holds for classical electrodynamics. This was called "Geometrodynamics" by Wheeler (who was famous for coining other phrases like blackhole and wormhole as well).



        Wheeler showed analytically that a properly constructed ball of gravitational and electromagnetic radiations would possess the properties of a massive object (he called this a geon) and he applied similar topological ideas such that electric charges can appear to exist when there is in fact none (a trick of spacetime geometry).



        To go further, one would need to be able to describe the gauge fields of the standard model in terms of geometry and properly quantize it. Whether these things can be found withing the topology of general relativity's spacetime, well that jury is still out.



        The equations involved here are highly nonlinear, especially when you have the feedback effect of describing say the electromagnetic field through geometry which in turn effects the geometry again.



        Anyway, when I read your question I am literally reading a book in front of my face:



        "The Geometrodynamics of Gauge Fields. On the geometry of Yang-Mills
        fields and Gravitational gauge theories"
        Eckehard W. Mielke



        I'll update this when I'm done maybe, it's an excellent book by the way.



        PS: I had the pleasure of seeing Kip Thorne Lecture this (last?) year and he seems a large proponent of geometrodynamics. As one of the founders of LIGO (we're FINALLY detecting gravitational waves!!! still can't believe it) there's a potential to test such theories in the foreseeable future.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 5 at 12:54

























        answered Dec 5 at 10:58









        R. Rankin

        1,252621




        1,252621























            5














            The answer is yes !



            In General Relativity, there's a not very well known theorem called the Campbell-Magaard theorem that states that any metric in 4D spacetime (including matter) could be represented (or embeded) as a metric in empty 5D spacetime (Ricci flat spacetime, wich implies pure geometry) :



            https://en.wikipedia.org/wiki/Campbell%27s_theorem_(geometry)



            https://arxiv.org/abs/gr-qc/0302015



            There are several simple solutions to Einstein's equation in 5D vacuum ($R_{AB}^{(5)} = 0$) that represent matter in 4D ($R_{mu nu}^{(4)} ne 0$). This could be interpreted as matter made of pure geometry, in higher dimensions spacetimes.



            Here's a very simple example. Consider the following metric in 5D spacetime ($theta$ is a cyclic coordinate in the fifth dimension) :
            begin{equation}tag{1}
            ds_{(5)}^2 = dt^2 - a^2(t) (dx^2 + dy^2 + dz^2) - b^2(t) , dtheta^2.
            end{equation}

            Substitute this metric into the 5D Einstein's equation without any matter :
            begin{equation}tag{2}
            R_{AB}^{(5)} = 0.
            end{equation}

            Then you get as a non-trivial solution these two scale factors :
            begin{align}tag{3}
            a(t) &= alpha , t^{1/2}, & b(t) &= beta , t^{- 1/2}.
            end{align}

            This is the same as pure radiation in an homogeneous 4D spacetime :
            begin{equation}tag{4}
            ds_{(4)}^2 = dt^2 - a^2(t) (dx^2 + dy^2 + dz^2).
            end{equation}

            With
            begin{equation}tag{5}
            R_{mu nu}^{(4)} - frac{1}{2} , g_{mu nu}^{(4)} , R^{(4)} = -, kappa , T_{mu nu}^{(4)},
            end{equation}

            and $T_{mu nu}^{(4)}$ describing a perfect fluid of incoherent radiation ($p_{rad} = frac{1}{3} , rho_{rad}$):



            begin{equation}tag{6}
            T_{mu nu}^{(4)} = (rho_{rad} + p_{rad}) , u_{mu}^{(4)} , u_{nu}^{(4)} - g_{mu nu}^{(4)} , p_{rad}.
            end{equation}



            This is the subject of the induced matter hypothesis, and is extremely fascinating! For more on the Campbell-Magaard theorem and the induced-matter theory :



            https://arxiv.org/abs/gr-qc/0507107



            https://arxiv.org/abs/gr-qc/0305066



            https://arxiv.org/abs/gr-qc/9907040






            share|cite|improve this answer























            • How are $rho_{rad}$ and $p_{rad}$ related to the geometric parameters $a(t)$ and $b(t)$?
              – lurscher
              Dec 5 at 14:25










            • @lurscher : $rho_{rad} propto a(t)^{- 4}$, which is the standard relation between energy density of radiation and the scale factor, in FLRW cosmologies.
              – Cham
              Dec 5 at 14:40










            • @Cham Is this a special 4d case of the Nash embedding theorem?
              – R. Rankin
              Dec 6 at 4:15










            • @R.Rankin, I don't know the Nash embedding theorem. Have you some references?
              – Cham
              Dec 6 at 12:12










            • Appears, Nash embedding is a case of whitney embedding. Here's a great stackexchange answer, but there are papers on them everywhere if your looking: math.stackexchange.com/questions/236285/…
              – R. Rankin
              Dec 6 at 12:30


















            5














            The answer is yes !



            In General Relativity, there's a not very well known theorem called the Campbell-Magaard theorem that states that any metric in 4D spacetime (including matter) could be represented (or embeded) as a metric in empty 5D spacetime (Ricci flat spacetime, wich implies pure geometry) :



            https://en.wikipedia.org/wiki/Campbell%27s_theorem_(geometry)



            https://arxiv.org/abs/gr-qc/0302015



            There are several simple solutions to Einstein's equation in 5D vacuum ($R_{AB}^{(5)} = 0$) that represent matter in 4D ($R_{mu nu}^{(4)} ne 0$). This could be interpreted as matter made of pure geometry, in higher dimensions spacetimes.



            Here's a very simple example. Consider the following metric in 5D spacetime ($theta$ is a cyclic coordinate in the fifth dimension) :
            begin{equation}tag{1}
            ds_{(5)}^2 = dt^2 - a^2(t) (dx^2 + dy^2 + dz^2) - b^2(t) , dtheta^2.
            end{equation}

            Substitute this metric into the 5D Einstein's equation without any matter :
            begin{equation}tag{2}
            R_{AB}^{(5)} = 0.
            end{equation}

            Then you get as a non-trivial solution these two scale factors :
            begin{align}tag{3}
            a(t) &= alpha , t^{1/2}, & b(t) &= beta , t^{- 1/2}.
            end{align}

            This is the same as pure radiation in an homogeneous 4D spacetime :
            begin{equation}tag{4}
            ds_{(4)}^2 = dt^2 - a^2(t) (dx^2 + dy^2 + dz^2).
            end{equation}

            With
            begin{equation}tag{5}
            R_{mu nu}^{(4)} - frac{1}{2} , g_{mu nu}^{(4)} , R^{(4)} = -, kappa , T_{mu nu}^{(4)},
            end{equation}

            and $T_{mu nu}^{(4)}$ describing a perfect fluid of incoherent radiation ($p_{rad} = frac{1}{3} , rho_{rad}$):



            begin{equation}tag{6}
            T_{mu nu}^{(4)} = (rho_{rad} + p_{rad}) , u_{mu}^{(4)} , u_{nu}^{(4)} - g_{mu nu}^{(4)} , p_{rad}.
            end{equation}



            This is the subject of the induced matter hypothesis, and is extremely fascinating! For more on the Campbell-Magaard theorem and the induced-matter theory :



            https://arxiv.org/abs/gr-qc/0507107



            https://arxiv.org/abs/gr-qc/0305066



            https://arxiv.org/abs/gr-qc/9907040






            share|cite|improve this answer























            • How are $rho_{rad}$ and $p_{rad}$ related to the geometric parameters $a(t)$ and $b(t)$?
              – lurscher
              Dec 5 at 14:25










            • @lurscher : $rho_{rad} propto a(t)^{- 4}$, which is the standard relation between energy density of radiation and the scale factor, in FLRW cosmologies.
              – Cham
              Dec 5 at 14:40










            • @Cham Is this a special 4d case of the Nash embedding theorem?
              – R. Rankin
              Dec 6 at 4:15










            • @R.Rankin, I don't know the Nash embedding theorem. Have you some references?
              – Cham
              Dec 6 at 12:12










            • Appears, Nash embedding is a case of whitney embedding. Here's a great stackexchange answer, but there are papers on them everywhere if your looking: math.stackexchange.com/questions/236285/…
              – R. Rankin
              Dec 6 at 12:30
















            5












            5








            5






            The answer is yes !



            In General Relativity, there's a not very well known theorem called the Campbell-Magaard theorem that states that any metric in 4D spacetime (including matter) could be represented (or embeded) as a metric in empty 5D spacetime (Ricci flat spacetime, wich implies pure geometry) :



            https://en.wikipedia.org/wiki/Campbell%27s_theorem_(geometry)



            https://arxiv.org/abs/gr-qc/0302015



            There are several simple solutions to Einstein's equation in 5D vacuum ($R_{AB}^{(5)} = 0$) that represent matter in 4D ($R_{mu nu}^{(4)} ne 0$). This could be interpreted as matter made of pure geometry, in higher dimensions spacetimes.



            Here's a very simple example. Consider the following metric in 5D spacetime ($theta$ is a cyclic coordinate in the fifth dimension) :
            begin{equation}tag{1}
            ds_{(5)}^2 = dt^2 - a^2(t) (dx^2 + dy^2 + dz^2) - b^2(t) , dtheta^2.
            end{equation}

            Substitute this metric into the 5D Einstein's equation without any matter :
            begin{equation}tag{2}
            R_{AB}^{(5)} = 0.
            end{equation}

            Then you get as a non-trivial solution these two scale factors :
            begin{align}tag{3}
            a(t) &= alpha , t^{1/2}, & b(t) &= beta , t^{- 1/2}.
            end{align}

            This is the same as pure radiation in an homogeneous 4D spacetime :
            begin{equation}tag{4}
            ds_{(4)}^2 = dt^2 - a^2(t) (dx^2 + dy^2 + dz^2).
            end{equation}

            With
            begin{equation}tag{5}
            R_{mu nu}^{(4)} - frac{1}{2} , g_{mu nu}^{(4)} , R^{(4)} = -, kappa , T_{mu nu}^{(4)},
            end{equation}

            and $T_{mu nu}^{(4)}$ describing a perfect fluid of incoherent radiation ($p_{rad} = frac{1}{3} , rho_{rad}$):



            begin{equation}tag{6}
            T_{mu nu}^{(4)} = (rho_{rad} + p_{rad}) , u_{mu}^{(4)} , u_{nu}^{(4)} - g_{mu nu}^{(4)} , p_{rad}.
            end{equation}



            This is the subject of the induced matter hypothesis, and is extremely fascinating! For more on the Campbell-Magaard theorem and the induced-matter theory :



            https://arxiv.org/abs/gr-qc/0507107



            https://arxiv.org/abs/gr-qc/0305066



            https://arxiv.org/abs/gr-qc/9907040






            share|cite|improve this answer














            The answer is yes !



            In General Relativity, there's a not very well known theorem called the Campbell-Magaard theorem that states that any metric in 4D spacetime (including matter) could be represented (or embeded) as a metric in empty 5D spacetime (Ricci flat spacetime, wich implies pure geometry) :



            https://en.wikipedia.org/wiki/Campbell%27s_theorem_(geometry)



            https://arxiv.org/abs/gr-qc/0302015



            There are several simple solutions to Einstein's equation in 5D vacuum ($R_{AB}^{(5)} = 0$) that represent matter in 4D ($R_{mu nu}^{(4)} ne 0$). This could be interpreted as matter made of pure geometry, in higher dimensions spacetimes.



            Here's a very simple example. Consider the following metric in 5D spacetime ($theta$ is a cyclic coordinate in the fifth dimension) :
            begin{equation}tag{1}
            ds_{(5)}^2 = dt^2 - a^2(t) (dx^2 + dy^2 + dz^2) - b^2(t) , dtheta^2.
            end{equation}

            Substitute this metric into the 5D Einstein's equation without any matter :
            begin{equation}tag{2}
            R_{AB}^{(5)} = 0.
            end{equation}

            Then you get as a non-trivial solution these two scale factors :
            begin{align}tag{3}
            a(t) &= alpha , t^{1/2}, & b(t) &= beta , t^{- 1/2}.
            end{align}

            This is the same as pure radiation in an homogeneous 4D spacetime :
            begin{equation}tag{4}
            ds_{(4)}^2 = dt^2 - a^2(t) (dx^2 + dy^2 + dz^2).
            end{equation}

            With
            begin{equation}tag{5}
            R_{mu nu}^{(4)} - frac{1}{2} , g_{mu nu}^{(4)} , R^{(4)} = -, kappa , T_{mu nu}^{(4)},
            end{equation}

            and $T_{mu nu}^{(4)}$ describing a perfect fluid of incoherent radiation ($p_{rad} = frac{1}{3} , rho_{rad}$):



            begin{equation}tag{6}
            T_{mu nu}^{(4)} = (rho_{rad} + p_{rad}) , u_{mu}^{(4)} , u_{nu}^{(4)} - g_{mu nu}^{(4)} , p_{rad}.
            end{equation}



            This is the subject of the induced matter hypothesis, and is extremely fascinating! For more on the Campbell-Magaard theorem and the induced-matter theory :



            https://arxiv.org/abs/gr-qc/0507107



            https://arxiv.org/abs/gr-qc/0305066



            https://arxiv.org/abs/gr-qc/9907040







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Dec 5 at 14:45

























            answered Dec 5 at 13:40









            Cham

            1,62111129




            1,62111129












            • How are $rho_{rad}$ and $p_{rad}$ related to the geometric parameters $a(t)$ and $b(t)$?
              – lurscher
              Dec 5 at 14:25










            • @lurscher : $rho_{rad} propto a(t)^{- 4}$, which is the standard relation between energy density of radiation and the scale factor, in FLRW cosmologies.
              – Cham
              Dec 5 at 14:40










            • @Cham Is this a special 4d case of the Nash embedding theorem?
              – R. Rankin
              Dec 6 at 4:15










            • @R.Rankin, I don't know the Nash embedding theorem. Have you some references?
              – Cham
              Dec 6 at 12:12










            • Appears, Nash embedding is a case of whitney embedding. Here's a great stackexchange answer, but there are papers on them everywhere if your looking: math.stackexchange.com/questions/236285/…
              – R. Rankin
              Dec 6 at 12:30




















            • How are $rho_{rad}$ and $p_{rad}$ related to the geometric parameters $a(t)$ and $b(t)$?
              – lurscher
              Dec 5 at 14:25










            • @lurscher : $rho_{rad} propto a(t)^{- 4}$, which is the standard relation between energy density of radiation and the scale factor, in FLRW cosmologies.
              – Cham
              Dec 5 at 14:40










            • @Cham Is this a special 4d case of the Nash embedding theorem?
              – R. Rankin
              Dec 6 at 4:15










            • @R.Rankin, I don't know the Nash embedding theorem. Have you some references?
              – Cham
              Dec 6 at 12:12










            • Appears, Nash embedding is a case of whitney embedding. Here's a great stackexchange answer, but there are papers on them everywhere if your looking: math.stackexchange.com/questions/236285/…
              – R. Rankin
              Dec 6 at 12:30


















            How are $rho_{rad}$ and $p_{rad}$ related to the geometric parameters $a(t)$ and $b(t)$?
            – lurscher
            Dec 5 at 14:25




            How are $rho_{rad}$ and $p_{rad}$ related to the geometric parameters $a(t)$ and $b(t)$?
            – lurscher
            Dec 5 at 14:25












            @lurscher : $rho_{rad} propto a(t)^{- 4}$, which is the standard relation between energy density of radiation and the scale factor, in FLRW cosmologies.
            – Cham
            Dec 5 at 14:40




            @lurscher : $rho_{rad} propto a(t)^{- 4}$, which is the standard relation between energy density of radiation and the scale factor, in FLRW cosmologies.
            – Cham
            Dec 5 at 14:40












            @Cham Is this a special 4d case of the Nash embedding theorem?
            – R. Rankin
            Dec 6 at 4:15




            @Cham Is this a special 4d case of the Nash embedding theorem?
            – R. Rankin
            Dec 6 at 4:15












            @R.Rankin, I don't know the Nash embedding theorem. Have you some references?
            – Cham
            Dec 6 at 12:12




            @R.Rankin, I don't know the Nash embedding theorem. Have you some references?
            – Cham
            Dec 6 at 12:12












            Appears, Nash embedding is a case of whitney embedding. Here's a great stackexchange answer, but there are papers on them everywhere if your looking: math.stackexchange.com/questions/236285/…
            – R. Rankin
            Dec 6 at 12:30






            Appears, Nash embedding is a case of whitney embedding. Here's a great stackexchange answer, but there are papers on them everywhere if your looking: math.stackexchange.com/questions/236285/…
            – R. Rankin
            Dec 6 at 12:30




















            draft saved

            draft discarded




















































            Thanks for contributing an answer to Physics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.





            Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


            Please pay close attention to the following guidance:


            • Please be sure to answer the question. Provide details and share your research!

            But avoid



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fphysics.stackexchange.com%2fquestions%2f445264%2fcan-matter-be-described-as-the-result-of-the-curvature-of-space-instead-of-vice%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            Popular posts from this blog

            Сан-Квентин

            8-я гвардейская общевойсковая армия

            Алькесар