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The hidden gravity
Enbang Li1
1
School of Physics, Faculty of Engineering and Information Sciences, University of Wollongong, Wollongong, NSW 2522 Australia
Newton’s law of universal gravitation states that the
attraction force between two point masses is directly
proportional to the product of their masses and inversely
proportional to the square of the distance between them
[1]. In the real world, no ideal point mass exists, but rather
all physical objects are of distributed masses. Therefore,
attentions need to be paid when Newton’s gravity law is
applied. According to Newton’s theorems on gravitation,
an ellipsoidally symmetric mass distribution can be
simplified to a point mass [2]. As a special case, a
spherical mass distribution satisfies the condition required
by the theorems, and also can be further treated as a point
source (with its total mass) located in its centre for both
near-field and far-field situations. However, for nonspherical mass distributions, this statement holds only for
the far-field and could become invalid for the near-field.
The variations of gravitational forces will become more
obvious with the increase of the oblateness or flatness of
the mass distribution. The extreme case is a disk-like mass
distribution, as shown in Fig.1c&d. In the vicinity of the
disk edge, 𝑘𝑔 can reach values which are much larger than
1, depending on the distance from the disk edge. For a case
where the distance is 0.01% of the disk radius (R), 𝑘𝑔
reaches 6. This means that a particle locating at that point
will experience a gravitational force which is more than 6
times stronger than that generated by a point source with
the same mass locating at the centre of the disk. In other
words, a disk of mass M acting on a particle at the disk
edge is equivalent to a spherically distributed mass of 6M.
More detailed calculations show that when the distance
between the particle and the disk edge is further reduced,
𝑘𝑔 will increase more rapidly.
Here we study the near-field gravitational forces generated
by non-spherical mass distributions for various shapes and
discover that the near-field gravitational force distributions
strongly depend on the mass distributions and can
significantly deviate from those generated by spherical
mass-distributions.
Galaxies all have very different shapes, from ellipticals,
spirals, barred spirals, to irregular galaxies, and their mass
distributions, in most cases, are complex and significantly
different from spherical. The additional gravitational
forces generated by non-spherical mass distributions, or
hidden gravity, could have significant implications for
kinematics of the cosmic structures.
.
Shown in Fig.1 are the calculated gravitational force
distributions for two different mass distributions: an oblate
spheroid and a disk. For comparison purpose, the
calculated gravitational forces are normalized by that
generated from a simplified point mass which is equivalent
to a spherical mass distribution. Plotted in Fig.1 are values
of a factor 𝑘𝑔 defined as 𝑘𝑔 = 𝐹/𝐹0 , where F is the
gravitational force between the test mass m and the
distributed mass with a total mass of M; and 𝐹0 is the
gravitational force between the test mass m and the
simplified point mass M. Therefore, for a non-spherical
mass distribution, 𝑘𝑔 represents how much the
gravitational force deviates from that generated by a
spherical mass-distribution.
It can be easily verified that for a spherical mass
distribution, 𝑘𝑔 is constantly unity in the space out of the
distributed mass, both in near- and far-field conditions.
For the oblate spheroid or revolving ellipsoid (the length
of semi-minor axis is half of that of semi-major axis, R)
mass distribution considered here, it is clear that near the
ellipsoid, 𝑘𝑔 increases to values larger than 1 along the
semi-major
axis
direction,
indicating increased
gravitational forces. Along the semi-minor axis direction,
we see that 𝑘𝑔 <1.
Figure 1: Calculated gravitational forces for two different
mass distributions
References
[1] Hartle, J. B. Gravity: an introduction to Einstein's
general relativity. (Addison-Wesley, 2003).
[2] Binney, J. & Tremaine, S. Galactic dynamics.
(Princeton University Press, 1987).
.
Path integral over causal spacetimes – a non-perturbative approach
E M Howard1
1Macquarie
University, Sydney, NSW, 2000, Australia
We attempt to describe the causal structure of quantum
spacetime as a path integral over geometries in a nonperturbative approach of quantum gravity theory. As we
know, gravity can’t be described as a renormalizable
quantum field theory within a four dimensional spacetime.
A perturbative expansion around a fixed background
geometry leads to a non-renormalizable theory.
The sum-over-histories picture (R. Feynman, J. Hartle)
[1] [2] [3] assigns probabilities to different sets of
alternatives at definite moments of time, instead of
associating quantum states on spacelike surfaces as in
Schrodinger-Heisenberg form. In order to keep the
“asymptotic safety”, an ultraviolet limit governing the
physics at distances shorter than the Planck scale must
exist and in its neighbourhood, the co-dimension of the
critical surface associated with it should be considered
finite. The infrared fixed limit is the ordinary classical
general relativity but also should possess a non-trivial
ultraviolet cut-off.
The complete dynamical nature of geometry, including
the diffeomorphism-invariant quantities in the continuum
are fully translated into a regularized causal theory. The
equivalent notion of observable from SchrodingerHeisenberg formulation is translated into diffeomorphism
invariant partitions of spacetime metrics. It is important
that the regularization should also restore the apparent
broken symmetries of the diffeomorphism invariance in
the continuum limit. In this formulation, instead of using
alternatives at definite moments of time, we use the more
general notion of spacetime alternatives, which is not
present in Schrodinger-Heisenberg form.
The new generalized spacetime formulation will be
completely free from the “problem of time”. We show that
causality and the non-local equations of motion can be
successfully combined in a picture of a quantum gravity in
which there is no well-defined notion of time. Nonlocality
and causality seem at first glance incompatible [4]. The
primary question is how causality and non-locality coexist
in a complete theory of quantum gravity while respecting
relativistic causality [5].
The employment of path integrals over spacetime
histories should help translating the dynamics associated to
unitary evolution into a generalized spacetime form. An
accurate sum-over-histories quantum gravity theory should
fully predict the probabilities associated with spacetime
alternatives while the notion of “measurement” and
“observable” should stop playing a fundamental role [6].
The key role in our approach is to separate the notions
of causality and causal order between events and the arrow
of time. The traditional view of causality is strictly
correlated to the unique direction from past to the future,
therefore to time’s arrow. All events are described as
causally ordered always according to time direction. We
suggest that the notion of time is at the origin of the
difficulties in formulating a quantum theory of gravity.
Based on a clear distinction between the notions of time
and causality [7], we search for a new formulation where
the quantum laws should be free from any temporal
correlations between operators. The causal structure of the
spacetime will remain fundamental to a quantum theory
but in a more generalized approach of sum-overspacetimes, while “the arrow of time” will be an emergent
property from quantum uncertainty but independent from
measurement. The longstanding Loschmidt paradox and
the origin of irreversible phenomena from reversible timesymmetric physical laws should also find resolution here.
The loss of information through quantum entanglement
plays the fundamental role here while reaching
thermodynamical equilibrium can be understood as a
fundamental property of all quantum systems.
Entanglement will continue to build up between the state
of a quantum system and the state of its external
environment, until equilibrium or a state of uniform energy
distribution is finally reached. Although there are deep
connections
between
quantum
mechanics
and
irreversibility, there is still an open question if quantum
mechanics is strictly required for irreversibility (including
why the universe is in a far from equilibrium state).
Causality, again, plays a significant role here, as it is
intrinsically connected to the non-local character of
quantum interactions [8]. A new non-perturbative
formulation of quantum gravity can be developed under
these new presumptions. As in our approach of quantum
gravity, entanglement is the essential ingredient that builds
the structure of space-time itself, we suggest that the
“arrow of time” emerges from the dynamical nature of
spacetime geometry, while transitioning from quantum to
classical and requiring a generalized spacetime
formulation of quantum mechanics.
References
[1] R.P. Feynman and A. Hibbs, Quantum Mechanics and
Path Integrals, McGraw-Hill, New York (1965).
[2] J.B. Hartle, Unitarity and Causality in Generalized
Quantum Mechanics for Non-Chronal Spacetimes, Phys.
Rev. D 49, 6543 (1994); quant-ph/9309012.
[3] J.B. Hartle, Spacetime Quantum Mechanics and the
Quantum Mechanics of Spacetime in Gravitation and
Quantizations, Les Houches Summer School Proceedings
Vol. LVII, Amsterdam (1995); gr-qc/9304006.
[4] Y. Aharonov, H. Pendleton, and A. Petersen, Int. J.
Theo. Phys. 2 (1969) 213; 3 (1970) 443; Y. Aharonov, in
Proc. Int. Symp. Foundations of Quantum Mechanics,
Tokyo, 1983, p. 10.
[5] S. Popescu and D. Rohrlich, Found. Phys. 24, 379
(1994).
[6] N. Yamada and S. Takagi, Prog. Theor. Phys. 87, 77
(1992).
[7] E Howard, Causal structure of general relativistic
spacetimes, AIP Conference Series, 1246 (2010); grqc/1601.06864.
[8] Č. Brukner, Bounding quantum correlations with
indefinite causal order, New J. Phys. 17, 073020 (2015).
Investigations of the shear-free
conjecture for a perfect fluid in general
relativity
P.A. Huf1 and J. Carminati1
1School
of Information Technology, Deakin University, Geelong,
Victoria 3216 Australia
1) The conjecture: The shear-free conjecture states that
if the shear of a perfect fluid is zero, then either the
expansion or rotation of the fluid must be zero.
σ=0 =>ωθ=0
It was initially formally stated by Treciokas & Ellis (1971)
[1], although there was some related discussion by Gödel
(1949) [2].
6) Role of software in algebraic investigations
The algebraic expressions in this conjecture are complex.
They can involve many equations including a large
number of terms (in tetrad form) or rigid algebraic
operations (in covariant formalism). Many research groups
employ software programs to assist with calculations. In
the current study we have created TensorPack, a new
covariant system which assists with abstract index
algebraic operations of tensors. The details of this can be
seen at [5].
7) Conclusion
The shear-free conjecture, if proven in the general case,
will provide additional insight into the Einstein field
equations. It can provide a workable comparison between
relativistic and Newtonian models.
2) Comparison to the Newtonian model: Importantly
this conjecture is in contrast to that predicted by the
Newtonian model, where there are known to be shear-free
fluids that are both rotating and expanding. The
contrasting kinematics of the two models has been
discussed in detail by Senovilla et al. [3].
3) Proofs and sub-cases of the conjecture: In general,
the conjecture has not been shown unconditionally, but has
been shown for the sub-cases [4]:
 dust (p=0) (Ellis, 1967) using tetrads; Senovilla et
al 2008 using covariant formalism; an alternative
covariant proof by Huf & Carminati
 Magnetic part of the Weyl tensor, H=0 (Collins,
1984)
 Electric part of the Weyl tensor E=0 (Carminati,
1988)
 Pure electromagnetic radiation p=(1/3)*mu
(Treciokas & Ellis, 1971)
 P=(-1/3)*mu (Cyganowski & Carminati, 2000)
 Mixture of matter and radiation: p=(-1/9)*mu (Van
den Bergh, 1999)
 Homogeneous spacetime (King & Ellis, 1973)
 Acceleration parallel to vorticity (Senovilla et al
2008)
For a review see [4].
4) Hence the motivation for the study involves:
 The relevance of the conjecture to the FLRW
cosmological model
 Analysis of why there is a difference between the
relativistic and Newtonian models
5) The general case is not yet solved, however there are
several research groups currently investigating further
subcases:
 The case where acceleration is orthogonal to
vorticity;
 The gamma and affine laws (where density is
linear function of energy density).
In general, these investigations involve covariant and/or
tetrad formalism.
Figure 1: The shear-free conjecture
References
[1] R.E. Treciokas, G.F.R. Ellis, Isotropic Solutions of the
Einstein-Boltzmann Equations, Commun. Math. Phys. 23
(1971) 1-22.
[2] K. Godel, Rev. An Example of a New Type of
Cosmological Solutions of Einstein's Field Equations of
Gravitation. Mod. Phys. 16, (1949) 2361
[3] J.M.M. Senovilla, C.F. Sopuerta, P. Szekeres, P.,
Theorems on shear-free perfect fluids with their
Newtonian analogues, Gen.Rel.Grav, 30 (1998) 389-411.
Updated (2008): https://archive.org/details/arxiv-grqc9702035
[4] H. R Karimian. Contributions to the study of shear-free
and of purely radiative perfect fluids in general relativity.
PhD thesis (2012)
[5] P.A.Huf, J.Carminati. TensorPack: a Maple-based
software package for the manipulation of algebraic
expressions of tensors in general relativity.
J.Phys.Conf.Ser. (2015) 633
Making Sense of Gravity
Peter R. Lamb1
1Institute
for Frontier Materials, Deakin University Waurn Ponds, VIC 3216 Australia
A fully relative theory of gravity (FRT) is outlined that
reproduces all the standard, observationally confirmed,
predictions of general relativity theory (GRT). However, it
avoids the need to hypothesise dark energy, dark matter and
cosmic inflation; explains the apparent absence of
antimatter; and is consistent with quantum mechanics and
the Standard Model of particle physics. Gravity still arises
from a distortion of space-time by the energy stored in
matter as mass. However, the distortion is a simple
expansion/contraction of space proportional to the product
of the speed of light and the time interval, and not a hidden
curvature in the underlying geometry. The result is that
magnitudes of physical laws have to be normalised by the
stored energy density but space is always flat. The theory
and observational consequences are set out more fully
elsewhere but give mathematically equivalent results when
examining small changes in a large background [1]. One
consequence is that the energy stored by particles,
oscillating states, decreases as the surrounding
(background) energy density increases. This leads to the
beautiful understanding that when an object falls in a
gravitational field some of its stored energy (mass) is
converted into the kinetic energy of motion. In contrast,
GRT hypothesises that the mass stays constant but the
kinetic energy comes from the surrounding field which is
then stronger, because all energy contributes to the field.
Under the rubber sheet analogy, in which masses distort
space-time leading to the apparent bending of light, GRT
has it that a surrounding uniform distribution of matter has
no effect on the distortion. It makes more sense to have a
central mass producing less distortion when there is more
surrounding matter. This is what is observed with a real
rubber sheet and, moreover, the speed of a wave
propagating on the sheet increases. FRT similarly has the
wave speed c increasing, but the energy stored as mass
obeying m  E / c 2 , with mass reducing as c increases.
Under GRT, the speed of light is constant and distortions
from matter in opposite directions in space cancel. This
cannot happen unless an expansion is matched by a
contraction, which is denied by symmetry. Under FRT, the
strength of gravitational interactions, length intervals, the
speed of light and the clock-rate are all dependent on the
background stored energy density. Thus, going back in
time, when the density of matter was larger because the
universe had expanded less, the clock-rate will have been
slower and the speed of light higher. The amount is just
right to explain the apparent faintness of distant supernovae
without the need to postulate dark energy (Figure 1). It also
allows now distant parts of the universe to have been
previously in equilibrium without hypothesising (faster than
the speed of light) inflation. The rotation curves of spiral
galaxies are also explained without the need for dark matter
if the background stored energy density tends to zero away
from the centre of spiral galaxies. This is possible if there
is an approximately uniform distribution of antimatter
galaxies and is allowed because FRT has it that matter and
antimatter, but not light, would be deflected from such
boundaries so that no annihilation signal would be seen. It
would be desirable to confirm that the gravitational lensing
of like-matter galaxy clusters can be fitted without the need
for dark matter. The equality of antimatter allows the
underlying physical laws to be symmetric with the apparent
asymmetry arising from the local excess of matter. There
are many other implications of FRT including that the event
horizons and singularities of dense concentrations of matter
(supposed black holes) cannot exist. This should already
have been appreciated, as it has been pointed out that the
idea that a photon loses energy in escaping a gravitational
field is mistaken [2-4]. The energy is unchanged, it is the
space-time of massive objects that is altered, and so light
cannot be trapped by gravity. Another implication is that
clock-rate must be increasing as the universe expands and
the amount accounts for most of the observed Pioneer
anomaly. The implications for particle physics are
profound and appear to fully support the Standard Model
with three flavour families and massless neutrinos that can
nevertheless oscillate, while yielding a revised
understanding of the Higgs mechanism and a prediction that
the Higgs boson mass should be mW  (mZ / 2) =
125.979±0.024 GeV/c2 [5], cf. 125.09±0.24 GeV/c2
measured [6].
Figure 1: Type 1a supernovae data [5] for luminosity distance
versus raw (Z) and corrected distance (Z(1+Z/2)).
References
[1] P. R. Lamb, A Fully Relative Theory of Gravitation,
version dated July 4, 2016:
http://dro.deakin.edu.au/view/DU:30054938.
[2] T.-P. Cheng, Relativity, Gravitation and Cosmology: A
Basic Introduction (OUP, 2009), 2nd ed. pp.77-111.
[3] L. B. Okun, K. G. Selivanov and V. L. Telegdi, Am. J.
Phys. 68, 115 (2000).
[4] J. Schwinger, Einstein's Legacy: The Unity of Space
and Time (Scientific American, New York, 1986), p. 142.
[5] J. Beringer et al. Phys. Rev. D 86, 1 (2012).
[6] G. Aad et al. Phys. Rev. Lett. 114, 191803 (2015).
[7] N. Suzuki et al., Astrophys. J. 746, 85 (2012).
Hovering vs Falling: Perspectives of black holes
Colin MacLaurin1
1
School of Mathematics and Physics, The University of Queensland, Brisbane, Qld, Australia
The properties of a black hole are typically interpreted using
Schwarzschild coordinates. In many ways, these correspond
to the measurements of observers hovering at fixed locations.
While this is a natural choice to make, it is certainly not the only
possibility. The measurements of observers freefalling radially
offer a contrasting perspective, and serve as a reminder that in
relativity space and time depend on the motion of the observer.
−1/2
dr is
For instance the textbook radial distance 1− 2M
r
1
what hovering observers measure, but this generalises to |e|
dr
for observers moving with arbitrary energy per mass e. As another example, the space part of spacetime may be represented
by a curved funnel known as “Flamm’s paraboloid”. This is
also based on the hovering perspective, however the falling observers measure space to be cone-shaped!
It is commonly stated that a rock falling into a black hole
will take a finite time as measured on its own watch, but that an
observer far away would determine an infinite time. However
simultaneity is relative, and this is based on a hovering perspective. Under the freefalling observers’ definition of simultaneity,
the rock and the distant observer record similar times for the
passage. Another popular description is that “time and space
swap roles inside the event horizon”. However there are popular coordinate systems for falling observers for which this is
not the case.
In conclusion, the contrasting perspectives of hovering
and falling observers clarify relativistic concepts and yield a
broader understanding of black holes.
Static vs Falling:
M
2.3
r=
Time slicings of
Schwarzschild black holes
Colin MacLaurin
www.ColinsCosmos.com
Introduction
Length-contraction (and expansion)
Popular descriptions of black holes often treat the Schwarzschild coordinate time slicing as
absolute, speaking of "the" distance, "the" space etc, with only limited qualification of
which observers are making these measurements. According to Eisenstaedt such a "neoNewtonian" interpretation lasted until the 1960s, when "black holes" replaced "frozen stars"
as the model of collapse. However vestiges of misinterpretation remain. We contrast the
perspectives of static and falling observers to emphasise the "relativity" of black holes.
A faller and static observer have relative 3-velocity given by the Lorentz factor:
The radial lengths obtained previously are factors of
and
relative to the static distance:
M
2.3
r=
Observer families and coordinates
M
2.3
r=
Suppose spacetime is filled with observers, all freefalling radially inward with the same
"energy per unit mass" e. All descriptions are a conglomeration of local measurements.
M
2.2
r=
2.2M
r=
e is invariant along a geodesic. Taylor & Wheeler use the metaphors rain/hail/drip,
however the
case is often forgotten, which I dub "mist":
1M
r=2.
M
r=
Choosing adapted coordinate systems is insightful:
Generalised Gullstrand-Painleve coordinates:
2.1
Faller perspective: both the static
observers and the distance between
them
are
increasingly
lengthcontracted as r decreases.
Static perspective: both the fallers and
the distance between them are
increasingly length-contracted.
Generalised Lemaitre coordinates:
"Space" part of spacetime
Use Schwarzschild coordinates for e=0 observers, since these are comoving (constant t).
Radial distance
The textbook radial "proper distance" is what static observers measure:
We can depict the curvature of space by a 2D surface embedded in Euclidean
same curvature. Take a constant "time" slice
and equatorial slice
cylindrical coordinates
the embedded surface z=z(r) has metric
with the
. In
Comparing, we obtain:
But for observers with arbitrary radial motion, this generalises to:
and
References
[1] R. Gautreau & R. Hoffmann (1978)
The slicing is not by Schwarzschild time, but adapted to the fallers. Also different
observers determine a different radial direction: though the above are both made by a
falling observer, they are measured in the faller and static radial 4-directions respectively.
Four equivalent approaches:
- adapted coordinates: set
- spatial projector:
To static observers, space is a funnel,
"Flamm's paraboloid".
- orthonormal tetrad frames:
- radar metric:
To the falling family, space is a cone for
|e|<1. For |e|=1 it is a flat plane, and for
|e|>1 it cannot be depicted by this method.
Time and space inside the event horizon
The Schwarzschild r-coordinate becomes timelike inside the event horizon. However this
must be defined by the hypersurface r=const, and not by the coordinate vector
which depends also on the other coordinates bundled with r:
[2] K. Martel & E. Poisson (2001), gr-qc/0001069
Coordinates
Inner product ∂r ·∂r = grr
1− 2M
r
Schwarzschild
[3] T. Finch (2015), gr-qc/1211.4337
The traditional perspective of infall
is what static observers measure,
at least somewhat.
The fallers' own perspective
("measurement") of their
infall is more natural.
3-Volume
The spatial volume inside the event horizon is relative to the motion of the observers
measuring it. Since the angular part of the metric is the Euclidean 2-sphere, and the
radial distance is proportional to r, the volume is proportional to the familiar Euclidean
volume as below. More formally, the volume element is:
generalised Gullstrand-Painlevé
generalised Lemaˆ
ıtre
Eddington-Finkelstein
(null version)
Eddington-Finkelstein
(timelikeversion)
−1
1
e2
1 2
(e −1+ 2M
r )
e2
Interpretation
spacelikefor r > 2M
timelikefor r < 2M
spacelike
spacelike
0
null
1+ 2M
r
spacelike
Though the coordinates r are identical (as maps
), the coordinate vectors are
not. In contrast, the surfaces r=const are independent of the other coordinates.
Hence the r-direction is spacelike everywhere for all observers except e=0, and so our
results relating it to length are justified. Clarity is needed with the common interpretations
that inside the horizon t and r swap roles, and that r decreases because it is timelike.
Conclusions and references
Radially moving observers determine interesting slicings of Schwarzschild spacetime.
Generalised Gullstrand-Painleve and Lemaitre coordinates are insightful about black holes.
which integrates to:
Finch shows this for e>0. Note raindrops (e=1) measure flat 3-space (Lemaitre 1933), as
do "mist rain" (e=-1) observers, hence both measure the Euclidean volume.
-
MacLaurin, C.; Davis, T.; Lewis, G., in preparation
Gautreau & Hoffmann (1978)
Taylor & Wheeler, Exploring Black Holes (2000)
Martel & Poisson (2001), gr-qc/0001069
Finch (2015), gr-qc/1211.4337v2
Figure 1: Existing version of poster
Dynamical angled brane
Kei-ichi Maeda1 and Kunihito Uzawa2
1
Department of Physics, Waseda University, Okubo 3-4-1, Shinjuku Tokyo 169-8555, Japan
Department of Physics, School of Science and Technology, Kwansei Gakuin University, Sanda, Hyogo 669-1337, Japan
2
We present the dynamical D-brane solutions describing
any number of D-branes whose relative orientations are
given by certain SU(2) rotations. These are the
generalization of the static angled D-brane solutions. We
study the collision of dynamical D3-brane with angles in
type II string theory, and show that the particular
orientation of the smeared D3-brane configuration can
provide an example of colliding branes if they have the
same charges. Otherwise a singularity appears before
D3-branes collide.
1) What is angled D-brane?
During the past two decades, understanding of
non-perturbative aspects of string theory has advanced
constantly. The development of these string theories has
brought with it the realization that extended objects such
as Dirichlet membranes (D-branes) beyond simply strings
play a crucial role. Another significant fact is that more
general D-brane solutions arise if the gravity is coupled
not only to single gauge field but to several combinations
of scalars and forms as intersecting brane solutions in the
string theory [1]. The intersections presented are actually
orthogonal in the sense that each D-brane lies along some
definite directions. It is however possible to show that
some configurations of several D-branes intersecting at
angles are still supersymmetric. Preserving supersymmetry
in such multiple D-brane configurations requires that the
angles are restricted to lie in an SU(N) subgroup of
rotations [2]. These static configurations may be useful in
trying to understand the microscopic counting of states of
four-dimensional black holes in string theories.
2) Motivation: “Why are angled D-branes studied?”
Although the time-dependent D-brane solutions in string
theory were introduced in [1, 3] and have been widely
used ever since, some aspects of the physical properties
such as dynamical D-brane oriented at angles and the
corresponding background field configurations remain
largely unexplored. Our motivation for the present work is
to improve this situation.
The static angled D-brane solutions was discovered by
[2] following the recognition of the several role of D-brane
configurations in string theory. The classical solution on
ten-dimensional spacetime that should be obtained to
introduce oriented at angles in string theories were studied
from several points of view. They have discussed the link
between the supersymmetric configurations, intersection of
brane and the rotation angles. The article [4] gives a
thorough review of much of what was known in the late
1990's. These results are more transparent if the dynamical
D-brane solution in string theory is discussed in terms of
supergravity theory, aiming to reduce everything to
ordinary higher-dimensional general relativity. In our
poster, we construct new solutions with nontrivial angles
in string theory and study the dynamical behaviour that
one would expect of a string theory with the
time-dependent fields and D-branes.
3) Our methods and results
We discuss the dynamical D-brane solution which shows
several D-branes oriented at angles with respect to one
another. Since the corresponding background field
configurations remain largely unclear, we present one such
class of solutions in the time-dependent D-brane
background. The dynamical solution which we construct in
this poster describes any number of D-brane whose
relative orientations are given by certain SU(2) rotations
[5]. These are functions of time that becomes static near
D-branes with the simplest possible dependence on the
warp factor. In the far region from angled D-brane in the
ten-dimensional background, the solutions give purely
contracting or expanding anisotropic universe.
We also provide more details about the collision of two
D3-branes (three-dimensional Dirichlet membranes) in the
presence of angles. The two D3-branes approach at an
angle in the direction of spatial part of world volume
coordinates. We have studied the dynamics of D3-branes
which have been smeared along the transverse space to
D3-branes. If two D3-brane charges are different from
each other, the distance between two D3-branes is still
finite when a singularity appears. Thus, we cannot describe
the collision of two D3-branes in terms of the solution. On
the other hand, there exists a complete collision where the
orientations of configurations between two smeared
D3-branes with same charges are either 0 or pi. This result
may be related to supersymmetry, which may be broken
for non-parallel branes.
4) Outlook for the future
Although the examples presented in our poster cannot
provide a realistic cosmological model, the solution may
be utilized to construct a cosmological solution just by
introduction of a test brane universe in higher-dimensions.
We may also construct new type of time-dependent black
hole solution with non-trivial angles by setting up more
complicated D-brane configuration. Those subjects are left
for a future works.
References
[1] K. Maeda, N. Ohta, K. Uzawa, JHEP 0906 (2009) 051
[2] J. C. Breckenridge, G. Michaud, R. C. Myers, Phys.
Rev. D 56 (1997) 5172
[3] G. W. Gibbons, H. Lu, C. N. Pope, Phys. Rev. Lett. 94
(2005) 131602
[4] D. Youm, Phys. Rept.316 (1999) 1
[5] K. Maeda, K. Uzawa, arXiv:1603.01948