Study of Regge Theory using 2-body motion in fractional space-time​
​by Utkarsh Patel​
​Indian Institute of Technology, Bhilai
Regge Theory in its usual form deals with the analytic properties of scattering amplitude as a function of angular momentum. The quantum mechanical treatment of potential in the theory leads to complex l-plane paths known as Regge trajectories. In its classical analog, for the bound state of an attractive spherically symmetric potential that falls into families with increasing angular momentum and energy, Regge’s idea expanded the phase space of momentum to include complex values of momentum as well. Based on a similar idea, this project aims to find the change in the equation of motion where particles are subjected to a fractional dimension universe. The most basic idea to work upon in such a scenario is to find the change in the orbital motion of a gravitating object given the fractional dimensions. We start with a potential form which is anything other than the usual inverse-square law and based on it, find the orbital equation. The idea could further be extended to solve the trajectories for a gravitating two-body system. In the case of general relativity, the precession of mercury’s orbit is a consequence of a 1/r^4 term in the force law. If such fractional dimensional calculations are done for such gravitating trajectories, the additional force law terms could be distant independent, which could possibly provide a solution for unusual galactic rotation curves at large distances. Apart from this, the project also aims to simulate the effective interactions among two bodies in fractional dimensions generated through the graph network approach in Wolfram Mathematica and compare the obtained trajectories with the classical approach.

Outline of the project:

◼
  • Structure of the work.
    • ◼
  • Basic Introduction to non-integer dimensions.
    • ◼
  • Motion of celestial bodies in fractional dimension space-time(numerical approach).
    • ◼
  • Approach of problem-at-hand using Wolfram Model.
    • ◼
  • Concluding Remarks.
    • ◼
  • Keywords.
    • ◼
  • Acknowledgement.
    • ◼
  • References.

    • 1. Structure of the work:

    The first section deals with the basic introduction of fractal dimensions and the fractional calculus. In the next section, we deal with the problem of two-body dynamics in a fractional space-time. We derive analytically the differential equation for the motion of bodies in the center of mass frame using the fractional calculus. After that, the obtained differential equation is solved numerically to simulate 2-body motion. The parameters of equation contains term responsible for the dimensionality of the space and the orbits of bodies are effected by it. In the last section, the problem at hand is tried using the hyper-graph theory, where the same motion is simulated on the hyper-graphs. The appendix section deals with the derivation of effective potential that a gravitating object experiences in a non-inertial reference frame. Based on the results, it is seen that the potential is added with an extra term dependent on the
    -3
    r
    , which in the next section can be seen as the cause of Mercury’s precession around sun(in GR, this effect is produced by the warping of space-time around a heavy mass like sun).

    2. Basic Introduction to non-integer dimensions:

    Fractals

    Mainstream mathematics in the past has been largely concerned with sets and functions to which the methods of classical calculus could be applied. This led researchers to consider non-sufficiently smooth or irregular functions as mostly pathological. They were considered more of individual curiosities rather than a branch for which a general theory could be developed. But lately, this trend has shifted and it has been realized that a great deal can be said about the mathematics of non-smooth objects. Moreover, it has been found that irregular sets provide a better representation of many natural phenomena than do the figures of classical geometry. Fractal geometry provide a general framework for studying methods of calculus. In loose terms, a fractal can be considered as a never-ending pattern. They are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. The object need not exhibit exactly the same structure at all scales, but the same “type” of structures must appear on all scales. Geometrically, they exist in between our familiar dimensions. A plot of the quantity on a log-log graph versus scale then gives a straight line, whose slope is said to be the fractal dimension[1]. Fractals patterns are found abundantly in nature, where common examples include coastlines, leaves pattern, hurricanes, seashells etc.
    A fractal pattern generated through Wolfram language using Fourier Series[1]:
    Out[]=
    ​
    animate
    show axes

    Fractional Calculus

    Fractional calculus could be said to begin with the seminal work of Leibniz (July 1, 1646 – November 14, 1716), where he used derivative without limits which could be generalized to discontinuous media like fractal derivative and q-derivative of quantum calculus. It is a generalization of the ordinary differentiation and integration to non-integer order. We may consider a continuous function, f(x), whose first-order derivative is defined as[2]:
    Df(x)=(df(x))/dx=f'(x)
    Now we define a new operator H as,
    D=
    1/α
    H
    ,α<1
    Then, we may wanna know that,
    Hf(x)=
    α
    D
    f(x)=?
    To reveal the basic idea of the fractional calculus in a layman way,we consider the following simplest equation[2]:
    y=x
    The first-order derivative is
    (1)
    y
    =1
    and the zeroth order derivative is defined as :
    (0)
    y
    =x
    Now, the question is what it is when order is α (0<α<1). Hereby,
    (α)
    y
    can be explained as the gradient of y on the fractal media, which might be continuous but not differentiable everywhere.

    3. Motion of celestial bodies in fractional dimension space-time(numerical approach):

    Introduction:

    The force experienced by a celestial body in orbit around a massive body has an extra potential term when the problem is dealt in a rotating non-inertial frame. This extra force term in the non-inertial frame is explained by the Coriolis effect due to rotation of the frame and is responsible for a extra term in the effective potential that the body in orbit experiences. This causes the orbits to precess slowly through their evolution in time. The analytical calculations for the same can be referred to from the Appendix section A.1. The effect is noticeable in an appreciable amount for the case of mercury’s orbit around sun as the mercury’s orbit is quite close to the sun. The same has been simulated in a scaled manner for a 2-plane of mercury’s orbit around sun in the Appendix section A.2.
    In this section for simulating the motion of a 2-body system in a fractional space-time, we start with a general form of force law for a fractional space, as given in reference[5] .
    ..
    r
    =2*
    1-
    D
    2
    π
    Γ
    D
    2
    *
    Gm
    2
    l
    0
    *
    1
    D-1
    (r/
    l
    0
    )
    (
    1
    )
    Here, D is the dimension of space-time and can have any real number value and
    l
    0
    is a scale factor(in meters) introduced to compensate for the dimensionality correctness of the equation. For simplicity sake, we take the value of scale factor to be 1.
    The equation (25) represents the acceleration experienced by a massive body due to the gravitational field of another massive body in a fractional dimension universe. For our problem in hand, we consider the case of two-body problem with one mass(m1) at radial distance r1 from origin and another mass(m2) at radial distance r2 in a center of mass frame. So, in that case the difference in radial position of the two bodies can be represented by a vector
    ->
    r
    given as:
    ->
    r
    1
    -
    ->
    r
    2
    . For the two bodies, the force law would be given by:
    m
    1
    ..
    r
    1
    =
    ->
    F
    ​​
    m
    2
    ..
    r
    2
    =-
    ->
    F
    Now, in center of mass frame:
    ->
    r
    =
    ->
    r
    1
    -
    ->
    r
    2
    ​​
    ->
    R
    =
    m
    1
    ->
    r
    1
    +
    m
    2
    ->
    r
    2
    m
    1
    +
    m
    2
    Here,
    ->
    R
    is the position vector of C.O.M. We may take it to be at origin for convenience and assume that C.O.M. has no initial velocity. Also, we know that in center of mass frame , the force on C.O.M. would be zero. So:
    ->
    ..
    R
    =0​​μ
    ->
    ..
    r
    =
    ->
    F
    Here, μ is the system’s reduced mass given as:
    μ=
    m
    1
    *
    m
    2
    m
    1
    +
    m
    2
    .
    Also, we see that as C.O.M. has no initial velocity, and the co-ordinate system is chosen in such a way that it lies at the origin, we have
    ->
    .
    R
    =0
    and
    ->
    R
    =0
    . So:
    ->
    r
    1
    =
    m
    2
    m
    1
    +
    m
    2
    ->
    r
    ​​
    ->
    r
    2
    =-
    m
    1
    m
    1
    +
    m
    2
    ->
    r
    (
    2
    )
    Now, as
    ->
    L
    (angular momentum) is a constant of motion in a central force system, so it implies that motion of the bodies is planar. So we choose the orientation of our co-ordinate system in such a way that our plane of motion happens to be on the X- and Y- axis. Then , we split the r vector into its x and y components and separately solve the x and y variable equations. Then, using the relation in (2), we can get the position vector functions of the two masses. The initial conditions used here are taken from the references[6,9] and they have been derived based on the references[10,11].

    Code:

    In[]:=
    Clear[t,x,y,e,m2]
    In[]:=
    ManipulateModule{rad},​​rad[t_,m2_,e_,α_]:=With{m1=1,k=1,l0=1,d=3+α},NDSolvex''[t]+2*
    1-
    d
    2
    π
    *Gamma[d/2](k/
    2
    l0
    )*x[t]l0
    d
    2
    (
    2
    (x[t]/l0)
    +
    2
    (y[t]/l0)
    )
    ==0,y''[t]+2
    1-
    d
    2
    π
    Gamma[d/2](k/
    2
    l0
    )*y[t]l0
    d
    2
    (
    2
    (x[t]/l0)
    +
    2
    (y[t]/l0)
    )
    ==0,x[0]==-1,x'[0]==0,y[0]==0,y'[0]==Sqrt[(1+e)],{x,y},{t,0,tp};Show[With[{m1=1},ParametricPlot3D[Evaluate[{{(m2/(m1+m2))*x[t],(m2/(m1+m2))*y[t],0}/.rad[t,m2,e,α],{-(m1/(m1+m2))*x[t],-(m1/(m1+m2))*y[t],0}/.rad[t,m2,e,α]}],{t,0,tp},PlotRange{{-3,3},{-3,3},(*Automatic,Automatic,*){-2.0,2.0}}(*,FrameTrue*),AxesFalse,ImageSize{550,540},BackgroundBlack]],With[{m1=1},Graphics3D[{Sphere[{Evaluate[{{(m2/(m1+m2))*x[tp],(m2/(m1+m2))*y[tp],0}/.rad[t,m2,e,α][[1]]}]},Min[.1*m2,.3]],Sphere[{Evaluate[{{(-m1/(m1+m2))*x[tp],(-m1/(m1+m2))*y[tp],0}/.rad[t,m2,e,α][[1]]}]},.1],Red,Point[{0,0,0}]},BackgroundBlack,BoxedFalse]]],{tp,0.1,"revolutions"},0.1,10*(2π)
    3
    2
    (1-e)
    ,ControlTypeTrigger(*Appearance"Labeled"*),{{m2,50,"M2/M1"},1,100,Appearance"Labeled"},{{e,0,"eccentricity"},0,0.9,Appearance"Labeled"},{{α,0.34,"fractional nature(3±α)"},-1,+1,0.0001,Appearance"Labeled"},ControlPlacementTop,SaveDefinitionsTrue

    Interpreting the results:

    The following image shows the snapshot taken for the simulation when the dimension of space time is taken to be 3.5 We see that the motion is bounded but on very large scales. Also, we see that the orbit of the massive body is continuously precessing about the center of mass. The time period for the loop to play is approx. 10000 seconds with initial velocity of massive object to be equal to 1 m/s and the other body at rest.
    The following image shows the snapshot taken for the simulation when the dimension of space time is taken to be 3.55 We see that the motion is not longer bounded and the interacting bodies fly off on their separate trajectories after a while. This transition of orbit being bounded or non-bounded with the increase of space-time dimensionality can be really interesting to study scattering theory. The time period for the loop to play is approx. 10000 seconds with initial velocity of massive object to be equal to 1 m/s and the other body at rest.

    4. Approach of problem-at-hand using Wolfram Model:

    Introduction:

    In this section, we attempt to simulate the 2-body motion on a hyper-graph of fractional dimensionality. As an initial step, we first create a 3D grid with cross links to increase its overall mean dimensionality to be somewhere between 3 to 4. Then, we choose a planar section of the grid to simulate 2-body motion.

    Code to construct non-integer dimensional grids:

    Here, the vertices number are changed with their co-ordinate location. This trick aids in simulating the motion and also in identifying grid points in terms of Cartesian co-ordinates.

    Grids with cross-links at Vertices:

    This here represents a 3D graph-grid with cross links at each node. A planar slice of this grid can be used to simulate motion of 2-body system.
    This is the 2-D graph grid which can be considered as a planar slice from 3-D grid.

    Dimensions of the Grid-graph:

    The created hyper-graphs with cross nodes for all the vertices can be checked for their mean dimensionality using the “WolframHausdorffDimension” resource function.

    Simulating 2-body motion on the hyper-graph:

    In order to simulate the motion of 2-body system on a fractional hypergraph, first we create 1) a graph-grid without cross nodes and then 2) a graph grid with cross nodes to form a space-time grid of integer and fractional dimensionality, respectively. Then the graph grid vertices are assigned the co-ordinate position depending in their location on the grid. So, the central most vertex is assigned a co-ordinate position of {0,0,0} or origin. The equation of motion for 2-body case is then numerically solved and the results obtained for each of the time value are fed in to graph co-ordinates iteratively and the vertex closest to the numerically obtained co-ordinates are then highlighted for geodesic balls with radii of 1. This simulates the motion of 2-body system over hypergraph.

    For the case when D=3

    The following function creates a grid graph stored in variable gv for our simulation.
    The following image shows a snapshot of 2-body motion on a graph, when the dimensionality of the space-time grid has been considered as D=3 . The two circles obtained depict the path of two massive bodies, where mass of heavier object is taken to be almost 1.5 times more than the lighter mass object. As the dimensionality of space-time grid is 3D, so we use inverse-square force law to obtain equations of motion and those EOMs are then solved and plotted on the graph-grid by selecting the nearest node to the co-ordinate points of solved equation. There is no precession seen in this case.

    For the case when D=3.1

    The following function creates a grid graph stored in variable gv for our simulation.
    The following image shows a snapshot of 2-body motion on a graph, when the dimensionality of the space-time grid has been considered as D=3.1 .The two circles obtained depict the path of two massive bodies, where mass of heavier object is taken to be almost 1.5 times more than the lighter mass object. As the dimensionality of space-time grid is 3D, so we use inverse-square force law to obtain equations of motion and those EOMs are then solved and plotted on the graph-grid by selecting the nearest node to the co-ordinate points of solved equation. There is precession seen in the motion of the bodies for this case.

    Comparison of the results obtained:

    5. Concluding Remarks:

    Main Results:

    The simulation of 2-body motion obtained through numerical solving of fractional dimension e.o.m. shows a precession in the orbit of masses around center of mass. The similar results have been obtained in literature through general relativistic calculations like in the case of mercury’s orbit. Thus, these result hint towards a possible connection of space-time warping with the variability in the dimension of space-time. Interestingly enough, we see that for dimensions above 3.6, the two-body orbits no longer remain bounded and the bodies fly apart into their separate trajectories. This could be taken as a reference to study scattering theory in higher dimensional motion. Also, a graph-grid with dimensionality in fractional order is created using Wolfram physics theory. This grid pattern consisting of cross-connected vertices serves as a space-time grid for 2-body motion. The results from both the approaches are compared by keeping the dimensionality of space-time same for both cases.

    Future Work:

    On the basis of analysis done in this project, the argument can be made that the effect of deviation in spatial dimension away from 3D can be incorporated into the effective potential term by introducing an extra potential term. Now we see that there is a lot of potential in this project for it to be extended after the completion of the school. Firstly, the equation of motion obtained through force-law can be compared with the Euler-Lagrange equations of motion obtained using Lagrangian mechanics. Also, after working on the problem through the approach of central forces, it would be interesting to extend it for the case of scattering theory. The effect of fractal space-time on the dynamics of Classical scattering theory can be understood well using the mathematical approaches followed during the project. A modification of Regge theory to incorporate effects of fractional space-time can also be aimed as a potential extension of the work.
    The project also included the simulation of 2-body problem on a fractional dimension grid of hypergraph with “WolframHausdorffDimension” greater than 3. Although the precession in motion was reproduced over hyper-graphs, the granularity of grid could be reduced to increase the overall resolution and correctness of the motion. The evolution of Runge-Lenz vector on a graph-grid with cross-links can be an interesting problem to study ahead.

    6. Keywords:

    ◼
  • Fractional Space-time.
    • ◼
  • 2-body problem.
    • ◼
  • Precession.
    • ◼
  • Hyper-graphs with cross-links.

    • 7. Acknowledgement:

    I am really grateful to Stephen Wolfram for giving me an opportunity to be a part of this summer school from Wolfram, also I thank him for selecting a really interesting project for me. I would like to thank my mentor Jon for his guidance and immense support throughout this project. He reviewed my work during all the crucial stages in these past three weeks. His humble and cheerful conduct motivated me a lot. Also, I would like to acknowledge my co-mentor Tom for his valuable contributions to my project. His knowledge and passion towards his work kept me motivated throughout. A big thanks to my fellow students Simon Fischer and Gianmarco Morbelli for all the time management in groups meeting, they had to do because of my time zone issue. I would also like to thanks Jonathan Gorard & Xerxes Arsiwalla for their crucial reviews during the mid-project meeting. At the last, I would like to thank Swastik Banerjee and all the other TAs for their assistance throughout the School.

    8. References:

    1
    .
    https://demonstrations.wolfram.com/FractalCurvesGeneratedByFourierSeries/
    2
    .
    https://link.springer.com/article/10.1007/s10773-014-2123-8
    3
    .
    https://en.wikipedia.org/wiki/Two-body_problem
    4
    .
    https://journals.sagepub.com/doi/abs/10.1177/1077546307077473?journalCode=jvcb
    5
    .
    https://arxiv.org/abs/2003.05784
    6
    .
    http://home.iiserb.ac.in/~sebastian/material/ClassMech/Week5_TwoBodyProblem_v1.pdf
    7
    .
    https://web.physics.ucsb.edu/~fratus/phys103/LN/TBP.pdf
    8
    .
    https://demonstrations.wolfram.com/MercurysPerihelionPrecession/#more
    9
    .
    https://www.youtube.com/watch?v=1i4dE3cp_ 44&list=LL&index=2&t=308s
    10
    .
    https://physics.stackexchange.com/questions/533517/solving-keplerian-two-body-problem-numerically
    11
    .
    https://en.wikipedia.org/wiki/Vis-viva_equation #:~:text=In %20 astrodynamics %2 C %20 the %20 vis %2 Dviva,object %20 is %20 its %20 own %20 weight.

    9. Appendix:

    A.2 Simulating Precession of Mercury’s orbit using EOMs from General Relativistic solution of 2-body problem:

    The observed precession in the orbit of the planet Mercury is 574.1 seconds of arc per century (relative to the International Celestial Reference Frame). According to Newtonian gravitational theory, taking account of the gravitational perturbations by the other planets (mainly Venus, Earth and Jupiter), a value of 531.6 is predicted. This anomaly of approximately 43 seconds of arc per century was first recognized by Le Verrier in 1859. The discrepancy was resolved by Einstein’s general theory of relativity in 1915.
    The general relativistic contribution can be approximated by the following model: “b” is the perturbation parameter given by:
    where, e is the eccentricity of the orbit
    The relativistic analogue of the orbit trajectory in the two-body problem is
    Initialization Cells:

    Main Code: