Membrane in M5-brane Background
Wei-shui Xua and Ding-fang Zengb
Institute of Theoretical Physics
P.O. Box 2735, Beijing 100080, P. R. China
College of Applied Science, Beijing University Of Technology
e-mail: wsxu@itp.ac.cn, dfzeng@bjut.edu.cn
Abstract
In this paper, we investigate the properties of a membrane in the M5-brane back-
ground. Through solving the classical equations of motion of the membrane, we can
understand the classical dynamics of the membrane in this background.
April 2007
http://arxiv.org/abs/0704.0577v3
1 Introduction
In eleven-dimensional M theory, there exists two extended brane solutions, i.e membrane
and M5-brane. The membrane was recovered in [1] as an elementary solution of D = 11
supergravity which preserves half of the spacetime supersymmetry, which is a electric
source of four-form field. While, the M5-brane was found in [2] as a soliton solution of
D = 11 supergravity also preserving half of the spacetime supersymmetry, but is magnetic
source of the same four-form field. These extended brane solutions can be related to the
corresponding brane solutions in ten-dimensional string theory. After performing the
compactification and some dualities, these branes can be reduced to D-branes or other
brane solutions in string theory [3].
In this paper, we will investigate the properties of M2-brane in the M5-brane back-
ground. Here, we will not investigate the cases of the brane intersection. Instead, we are
mainly concerned with the classical dynamics of membrane in the given background. As
will be illustrated, due to the gravity force of M5-brane, the membrane evolves nontriv-
ially.
In the 11-dimensional supergravity, the classical solution of N coincident M5-brane
reads
ds2 = H−
3 ηµνdx
µdxν +H
3 δijdx
idxj,
H = 1 +
πNl3p
(xi)2 = r2 + x11
2, µ, ν = 0, 1, · · · , 5, i, j = 6, 7, 8, 9, 11 (1.1)
and the 4-form field strength takes the form
F4 = dA3 = 3πNl
pdvS4 (1.2)
where the dvS4 denotes the volume form of a unit S
4 and lp is the Planck length in the
11-dimensional theory. The N coincident M5-brane are parallel to the xµ directions and
located at R = 0 in the transverse space. In the near horizon limit R → 0, the harmonic
form H will become H =
πNl3p
, and the other parts will choose the same forms as in the
equations (1.1) and (1.2).
As in [4], if we suppose that there are a periodic configuration of N coincident M5-
brane along the x11 direction at intervals of 2πR11, and take the limit of 1 ≪ r/R11, then
our background metric and the 4-form field strength will become
ds2 = f−
3ηµνdx
µdxν + f
3 δijdx
idxj + f
3 (dx11)2,
f = 1 +
R11r2
2Nℓ3p
dvS3 ∧ dx11,
(xi)2, x11 = R11φ, (1.3)
where µ, ν = 0, 1, · · · , 5, i, j = 6, 7, 8, 9 and 0 ≤ φ ≤ 2π . We can see this metric has an
so(4) symmetry group of rotations in the directions transverse to the M5-brane. In the
near horizon limit, the harmonic function f becomes f =
R11r2
. While, the other parts
of background (1.3) remain unchanged. Actually, if letting the radius of x11 coordinate
approach zero, then the metric (1.3) can reduce to the N coincident NS5-brane solution
in ten-dimensional string theory [5].
Here we will mainly focus on the classical dynamics of a M2-brane in the above back-
grounds (1.1) and (1.3). The dynamics of this single membrane can be described by the
Nambu-Goto and Wess-Zumino type effective action. However, for the coincident mem-
branes, unlike the coincident D-brane in string theory which can be described by the
effective action [6], their worldvolume action is still not very clear [7]. We choose the
worldvolume coordinates of membrane as x0, x1, x2, and those of M5-brane as x0, · · · , x5.
Hence M2-brane is “parallel” to the M5-brane, i.e it is extended in some of the M5-brane
worldvolume directions xµ, and point-like in the directions transverse to the M5-brane
(x6, x7, x8, x9, x11). Indeed, this configuration breaks supersymmetry completely. We can
label the worldvolume coordinates of the M2-brane by ξµ, µ = 0, 1, 2, and use reparame-
terization invariance on the worldvolume of the M2-brane to set ξµ = xµ. The position of
the M2-brane in the transverse directions, (x6, · · · , x9, x11), give rise to scalar fields on the
worldvolume of the M2-brane, (X6(ξµ), · · · , X9(ξµ), X11(ξµ)). A single M2-brane world-
volume action [8] is given by the sum of the Nambu-Goto action and the Wess-Zumino
type term in the following form
SM2 = −T2
− detP [G]µν + T2
P [A] (1.4)
where the tension of the M2-brane is expressed as T2 = 1/4π
2l3p, and P [· · ·] means the
pullback operation
P [G]µν =
GMN(X),
P [A] =
AMNL(X) . (1.5)
The indices M,N,L run over the whole eleven dimensional spacetime. And the fields
GMN , AMNL denote the metric and form field in eleven dimensions. In the following
sections, we will discuss the M2-brane classical dynamics in the above backgrounds, and
suppose that the transverse coordinates of M5-brane only depend on the time coordinate.
In this case the Wess-Zumino term in the membrane action will vanish.
2 Classical dynamics of membrane
Now let us consider the membrane dynamics in the background (1.1). Since we have
supposed that the directions transverse to the M5-brane X i are only the function of time
t, where i = 6, 7, 8, 9, 11, the pullback quantities are as following
P [G]tt = −H−
3 Ẋ iẊ i,
P [G]x1x1 = H
P [G]x2x2 = H
3 , P [A] = 0. (2.1)
after substituting the above equations (2.1) into the M2-brane action (1.4), we get
SM2 = −V T2
H−1 − Ẋ iẊ i (2.2)
where V is the space volume of the M2-brane, also i = 6, · · · , 9, 11. We can find it is very
similar to the corresponding one in [9] which is the DBI action of D-brane in the N NS5
brane background. Then through using the Legendre transformation, the Hamiltonian is
H−1 − Ẋ iẊ i
≡ V E (2.3)
where the E denotes the energy density. And the equation of motion will be
H−1 − ẊjẊj
H−1 − ẊjẊj
. (2.4)
Using this equation of motion (2.4), one can check that the Hamiltonian is conserved. To
solve the (2.4), we need the initial conditions that it is ~X(t = 0) and ~̇X(t = 0). These
two vectors define a plane in R5. By an SO(5) rotation, we can rotate this plane into
the (x6, x7) plane. Then the motion will remain in the (x6, x7) space for all time. Thus,
without loss of generality, we can study trajectories in this space.
We choose the polar coordinates
X6 = R cos θ,
X7 = R sin θ. (2.5)
Then the energy density (2.3) will become
H−1 − Ṙ2 − R2θ̇2
, (2.6)
and the angular momentum density will be
H−1 − Ṙ2 − R2θ̇2
. (2.7)
We can find this angular momentum of the M2-brane is conserved as well. From the
membrane action (2.2), we can obtain energy momentum tensor. The components of Tµν
are listed in the following
T00 = −
H−1 − Ẋ iẊ i
Tij = −T2δij
H−1 − Ẋ iẊ i, (2.8)
and the other components of stress tensor are zero. From the angular momentum Lθ
equation (2.7) and energy density E (2.6), we can get the equations of the coordinates R
and θ
Ṙ2 =
T 22 +
, (2.9)
EH(R)R2
. (2.10)
For simplicity, we can first consider Lθ = 0 case, then the radial equation is
Ṙ2 =
. (2.11)
The right hand of the above equation can’t be smaller than zero, so we get a constraint
on the coordinate R is
πNl3p
− 1. (2.12)
From the above equation, we can see if the energy density E is larger than the tension of
a M2-brane, T2, the constraint (2.12) is empty and the M2-brane can escape to infinity.
However, for E < T2, the M2-brane does not have enough energy to overcome the grav-
itational pull of the M5-brane, and then will fall down to the M5-brane from an initial
position.
Choosing the near horizon limit, hence the harmonic function becomes H = πNl3p/R
Then the equation (2.11) will be
Ṙ2 =
πNl3p
π2N2E2l6p
R6. (2.13)
Since the left hand of the equation (2.13) is nonnegative, the coordinate R has a maximal
value
πNE2l3p/T
. Also from this equation, the minimal value of R is zero. Except
for these two, there are no other extremum. But there is one inflexion between points
R = 0 and
πNE2l3p/T
. We can regard the M2-brane is at the maximal value
πNE2l3p/T
at the initial time. Due to the gravitational force of M5-brane, the M2-
brane then will roll down to the M5-brane. As the time t → ∞, the radial coordinate
R approaches to zero. We can calculate the energy momentum tensor which is Tij =
EH(R)
as the R → 0, the Tij will approach to zero. It may regard as the pressure
decreasing to zero. But we need to mention that the coordinate R can’t reach zero, since
at this point the supergravity background will be not reliable. Then the classical dynamics
of the membrane near R = 0 from the above analysis will become incorrect. Thus, in
order to use the supergravity approximation, we must constrain the coordinate R to be
larger than the planck length lp.
Now we begin to consider the nonzero case of angular momentum. From the radial
equation of motion (2.9), and after substituting the harmonic function H = 1+πNl3p/R
we can get the constraint on the radial coordinate R is
πNE2l3p
πNl3p
. (2.14)
If choosing the equal case of the above equation, the constraint will become
πNE2l3p
πNl3p
= 0. (2.15)
The above equation only has one real root which is the maximal distance that M2-brane
is separated from M5-brane.
For simplicity, we choose the near horizon limit, then the equation of motion for the
radial coordinate will become
Ṙ2 =
πNl3p
π2N2E2l6p
π2N2E2l6p
R6. (2.16)
We find that the equation (2.16) is still very difficult to solve. Instead, here, we take
some analysis for this equation. If letting Lθ = 0, then this equation will reduce to the
equation (2.13). We let the left hand of the equation (2.16) to zero, then we can get the
extremal value for R. Actually, there are two only two real extremal values of the radial
coordinate R. One is R = 0, the other is
108πNlp
3E2T2 + 12
6 + 27π2N2lp
6E4T2
− 12Lθ2
108π Nlp
3E2T2 + 12
6 + 27π2N2lp
6E4T2
. (2.17)
When Lθ = 0, the above R value will reach the
πNE2l3p/T
. As the same in the
Lθ = 0 case, between the R = 0 and (2.17) there exists a inflexion. We can suppose
that the M2-brane is at the maximal value (2.17) at the initial time, then under the
gravitational pull of M5-brane, it will monotonic approach to M5-brane. Of course for the
Lθ nonzero case, the equation (2.10) for the θ coordinate in the near horizon background is
θ̇ = Lθ
= RLθ
πNEl3p
. Thus, if the radial coordinate R reaches the value (2.17), the angular
velocity will choose the maximum, and as the R → 0, the angular velocity does also
approach to zero. The energy momentum tensor satisfies Tij = −δij
EH(R)
= −δij
πNEl3p
Thus, it again goes to zero as in the Lθ = 0 case. As mentioned in the above, near the
region R = 0, the classical background will be instability due to the strong interaction.
Hence the above supergravity analysis will become unreliable in this region.
From the first section, we already know that, after compactifying a periodic circle of
coordinate x11, the metric (1.1) will become background (1.3). In the following, we study
the membrane dynamics in this background (1.3). Here, we still suppose the directions
transverse to the M5-brane X i and X11 are only the function of time t, where i = 6, 7, 8, 9,
then the pullback quantities take the form as follows
P [G]tt = −f−
3 + f
3 Ẋ iẊ i + f
3 Ẋ11Ẋ11,
P [G]x1x1 = f
P [G]x2x2 = f
3 , P [A] = 0. (2.18)
After inserting (2.18) into the M2-brane action (1.4), we can get
SM2 = −V T2
f−1 − Ẋ iẊ i − Ẋ11Ẋ11 (2.19)
where V is the space volume of the M2-brane. This action is also very similar to action
in [9] except for the harmonic function and dimension. From the Lagrangian (2.19), we
can derive the equations of motion for the membrane in this background as followes
f−1 − ẊjẊj − R211φ̇2
f−1 − ẊjẊj −R211φ̇2
, (2.20)
R11φ̇
f−1 − ẊjẊj − R211φ̇2
 = 0. (2.21)
Due to some symmetry of this system, there are also some conserved charges. Time
translation invariance implies that the energy
H = PiẊ i + Pφφ̇− L (2.22)
is conserved. The momentum is obtained by varying the Lagrangian L,
δẊ i
T2V Ẋi
f−1 − ẊjẊj − R211φ̇2
, (2.23)
T2V R
f−1 − ẊjẊj − R211φ̇2
. (2.24)
Substituting (2.23) into (2.22), we find that the energy is given by
f−1 − Ẋ iẊ i − R211φ̇2
≡ V E. (2.25)
And since the harmonic function f = 1 +
R11r2
, then ∂if(r) = X
if ′(r)/r, and one of the
equations of motion (2.20) can be rewritten as
f−1 − ẊjẊj − R211φ̇2
X if ′
2rf 2
f−1 − ẊjẊj − R211φ̇2
, (2.26)
the other one is unchanged.
To solve these equations, we need to specify some initial conditions for the coordinates.
One condition is ~X(t = 0) and ~̇X(t = 0). These two vectors define a plane in R4. By an
SO(4) rotation symmetry, we can rotate this plane into the (x6, x7) plane. The other one
is φ(t = 0) and φ̇(t = 0). Then the motion of the membrane will remain in the (x6, x7, φ)
space for all time. Thus, without loss of generality, we can study trajectories in this space.
In addition to the energy, the angular momentum of the M2-brane is conserved as well.
It is given by
(X6P 7 −X7P 6). (2.27)
Using the expression for the momentum, (2.23), we find that
Lθ = T2
X6Ẋ7 −X7Ẋ6
f−1 − ẊjẊj −R211φ̇2
. (2.28)
Another interest quantity is the stress tensor Tµν associated with the moving M2-
brane. The component T00 denotes the energy density, so it is given by expression (2.25)
for E, with the factor of the volume stripped off. We list the components of Tµν in the
following equations
T00 = −
f−1 − Ẋ iẊ i −R211φ̇2
Tij = −T2δij
f−1 − Ẋ iẊ i − R211φ̇2,
Tφφ = −T2R211
f−1 − Ẋ iẊ i −R211φ̇2, (2.29)
and the other components of stress tensor are zero.
Due to the so(4) rotation symmetry in the transverse directions of M5-brane, it is
convenient to change to the polar coordinates
X6 = r cos θ, X7 = r sin θ. (2.30)
In these coordinates, the expressions of the energy density and angular momentum density
becomes
f−1 − ṙ2 − r2θ̇2 − R211φ̇2
, (2.31)
f−1 − ṙ2 − r2θ̇2 − R211φ̇2
, (2.32)
f−1 − ṙ2 − r2θ̇2 − R211φ̇2
. (2.33)
One can check directly that Lθ and Lφ are conserved by using the equations of motion
(2.26) and (2.21).
In order to solve the equations of motion for the given energy and angular momentum
densities E, Lθ and Lφ, we would like to solve the equation (2.32) for θ̇, and then substitute
this solution into the (2.31). Then the equation for the θ̇ is
. (2.34)
Inserting it into (2.31), (2.32) and solving for ṙ, we find
ṙ2 =
E2f 2
T 22 +
. (2.35)
Also we have the equation of φ̇
EfR211
. (2.36)
In the next, we would like to study the solutions of the equations of motion (2.34), (2.35)
and (2.36).
Firstly, we consider the angular momentum Lθ = 0 case. Then Equation (2.34) implies
that θ is constant, while the radial equation (2.35) takes the form
ṙ2 =
E2f 2
T 22 +
. (2.37)
Since the right hand side of the equation(2.37) is non-negative, then we can get the
condition 1
T 22 +
≥ 0. After substituting the harmonic function f , (1.3),
into it, we find the constraint on r (for fixed energy density E)
R11r2
− 1 (2.38)
where we can define the effective M2-brane tension is
T 2e = T
(2.39)
From the equation of constraint (2.38), obviously, if the energy density E is larger than
the effective tension of a M2-brane, Te, the constraint (2.38) is empty and the M2-brane
can escape to infinity. For E < Te, the M2-brane does not have enough energy to escape
the gravitational pull of the M5-brane, which means that it cannot exceed some maximal
distance from the M5-brane.
Under the near horizon limit, the harmonic function f will become f =
R11r2
. Then
the equation (2.38) will be
R11r2
. Thus, if r <<
, the effective tension of
membrane Te satisfies the constraint Te/E >> 1. However, r >>
, the case will
be otherwise. Indeed, in this near horizon case, we can solve for the trajectory r(t), φ(t)
exactly. Substituting the harmonic function f =
R11r2
into (2.37), we find the equation
of motion
ṙ2 =
E2N2l6p
T 22 +
r4. (2.40)
Then the solution can be obtained
L2φ +R
NR11E2l3p
t (2.41)
where we choose t = 0 to be the time at which the M2-brane reaches its maximal distance
from the M5-brane. For an observer living on M5-brane, the M2-brane reaching r = 0
will take an infinite time. Also, the M2 radial motion is similar to D-brane’s motion in
And the equation of motion (2.24) becomes
φ̇2 =
r2 − ṙ2
11 + T
. (2.42)
Substituting the solution r into equation (2.42), we can get the equation
L2φ + T
. (2.43)
Then after solving this equation, the solution can be obtained
L2φ + T
t. (2.44)
It is interesting to calculate the energy momentum tensor of the M2-brane in this
case. The energy density T00 is constant and equal to E throughout the time evolution.
However, for the parts Tij and Tφφ, we can find
Tij = −δij
Tφφ = −
R211T
. (2.45)
We see that the pressure goes smoothly to zero as r → 0, since f(r) ∼ 1/r2. But again
as the analysis in the background (1.1), this may be unreliable near the r = 0 region.
So far we have discussed the trajectories with vanishing angular momentum density
(2.32). A natural question is whether anything qualitatively new occurs for non-zero Lθ.
Just as [9], we can think as follows, the radial equation of motion (2.35) can be thought
of as describing a particle with mass m = 2, moving in one dimension r in the effective
potential
Veff(r) =
E2f 2
T 22 +
(2.46)
with zero energy. Now we discuss the properties of this effective potential Veff . In the
small r region, it will behave as
Veff(r) ≃
E2Nl3p
r2. (2.47)
For large r, the leading terms of this potential will be
Veff(r) ≃
− 1. (2.48)
If the energy density of the M2-brane is smaller than the effective tension of a M2-brane,
E < Te, then the effective potential Veff approaches to a positive constant (2.48) as r → ∞,
which means the membrane cannot escape to infinity. From the equation (2.47), we can
find that in order to have trajectories at non-zero r, the angular momentum must satisfy
the constraint
NEl3p
. (2.49)
If the constraint (2.49) is not satisfied, the only solution is r = 0. But, if the condition
(2.49) is satisfied, the trajectory of the M2-brane is qualitatively similar to that in the
Lθ = 0 case. It will approach the M5-brane and does not have stable orbits at finite r.
For the case Te >> E, the whole trajectory lies again in the region r <<
Nl3p/R11,
and one can approximate the harmonic function (1.3) by f =
R11r2
. Then the equation
(2.35) for ṙ will be
ṙ2 =
Nl3pE
E2N2l4p
T 22 +
r4 , (2.50)
with the solution
L2φ +R
R11NE2l3p − R11L2θ
NR11E2l3p − R211L2θ
NEl3p
t. (2.51)
We can find that the non-zero angular momentum can slow down the exponential decrease
of r as t → ∞. In the near horizon limit f(r) = Nl
, the solution of the equation (2.34)
for θ is
R11Lθ
ENl3p
t. (2.52)
The solution (2.51) and (2.52) mean that the M2-brane in the background (1.3) will be
spiralling towards the origin, circling around it an infinite number of times in the process.
The equation about φ is
Lφ(NE
2l3p −R11L2θ)
ENl3p(L
cosh
R11E2Nl3p − R211L2θ
ENl3p
. (2.53)
and the solution of the above equation reads
L2φ +R
NE2l3p − L2θ
R11NE2l3p − R211L2θ
ENl3p
t. (2.54)
At t = 0, the φ = 0, however, the time t → ∞, then, φ → Lφ
NE2l3p−L
Thus, the non-zero angular momentum Lθ slows down the variation of φ. From these
three solutions, we know that the M2-brane is circling along the θ direction, varying along
the φ and falling down towards the M5-brane in the process. Also, the energy momentum
tensor Tij and Tφφ will approach to zero as r → 0, since f(r) ∼ 1/r2. But we must
mention that, near the r = 0 region, the discussion may be incorrect due to the strong
coupling.
In the background (1.3), the results about the dynamics of a M2-brane have some
similar properties as studying in [9]. This can be understood that the D2-brane and
NS5-brane in IIA can be got by compactified one transverse dimension of M2-brane and
M5-brane in M theory. The solutions of equation of motion describe the M2-brane falling
towards the M5-brane. In the non-zero angular momentum Lθ, the M2-brane is spiralling
towards the M5-brane. But both in this two case, M2-brane has a angular momentum Lφ.
We need to mention that the background (1.3) is only correct in the limit of 1 ≪ r/R11.
Therefore, as the M2-brane approaches the M5-brane, the energy momentum tensor Tij
and Tφφ approaching zero may be unreliable. Since here the radial coordinate r is smaller
than the radius R11. So we are not sure whether the membrane will have the same
behavior just like the late time behavior of unstable D-brane [10, 11, 12, 13, 14].
In the above sections, we investigated the membrane classical dynamics in various
M5-brane backgrounds. There may be some generalizations, since under the Penrose
limit, the N coincident M5-brane solution (1.1) will reduce to the AdS7 × S4 geometry.
Hence one can investigate the membrane dynamics in this geometry. For the (1.1), (1.3)
and their near horizon background geometry, after calculating the classical equations of
motion of membrane from the membrane action (1.4), we can analyze the moving trajec-
tories of membrane. In some particular cases, we can get the exact solution of trajectories
of membrane. However, generally, the equations of motion is very difficult to solve. But
through analyzing these equations, we still can obtain some qualitative information about
the motion of membrane. Consequently, in the M5-brane background, the membrane will
be falling and spiralling towards to the M5-brane by the gravitational force of M5-brane.
In the near M5-brane region, i.e R (or r) being of the order of the planck length lp,
the above analysis of the classical dynamics of membrane may not be trusted, since the
method of the supergravity approximation is unreliable.
Acknowledgements
We would like to thank Yi-hong Gao for the useful suggestions and discussions.
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Phys. B647, 101 (2002) [hep-th/0208196].
[14] N. Lambert, H. Liu and J. Maldacena, “Closed strings from decaying D-branes,”
hep-th/0303139.
	Introduction
	Classical dynamics of membrane

  In this paper, we investigate the properties of a membrane in the M5-brane
background. Through solving the classical equations of motion of the membrane,
we can understand the classical dynamics of the membrane in this background.

Introduction
In eleven-dimensional M theory, there exists two extended brane solutions, i.e membrane
and M5-brane. The membrane was recovered in [1] as an elementary solution of D = 11
supergravity which preserves half of the spacetime supersymmetry, which is a electric
source of four-form field. While, the M5-brane was found in [2] as a soliton solution of
D = 11 supergravity also preserving half of the spacetime supersymmetry, but is magnetic
source of the same four-form field. These extended brane solutions can be related to the
corresponding brane solutions in ten-dimensional string theory. After performing the
compactification and some dualities, these branes can be reduced to D-branes or other
brane solutions in string theory [3].
In this paper, we will investigate the properties of M2-brane in the M5-brane back-
ground. Here, we will not investigate the cases of the brane intersection. Instead, we are
mainly concerned with the classical dynamics of membrane in the given background. As
will be illustrated, due to the gravity force of M5-brane, the membrane evolves nontriv-
ially.
In the 11-dimensional supergravity, the classical solution of N coincident M5-brane
reads
ds2 = H−
3 ηµνdx
µdxν +H
3 δijdx
idxj,
H = 1 +
πNl3p
(xi)2 = r2 + x11
2, µ, ν = 0, 1, · · · , 5, i, j = 6, 7, 8, 9, 11 (1.1)
and the 4-form field strength takes the form
F4 = dA3 = 3πNl
pdvS4 (1.2)
where the dvS4 denotes the volume form of a unit S
4 and lp is the Planck length in the
11-dimensional theory. The N coincident M5-brane are parallel to the xµ directions and
located at R = 0 in the transverse space. In the near horizon limit R → 0, the harmonic
form H will become H =
πNl3p
, and the other parts will choose the same forms as in the
equations (1.1) and (1.2).
As in [4], if we suppose that there are a periodic configuration of N coincident M5-
brane along the x11 direction at intervals of 2πR11, and take the limit of 1 ≪ r/R11, then
our background metric and the 4-form field strength will become
ds2 = f−
3ηµνdx
µdxν + f
3 δijdx
idxj + f
3 (dx11)2,
f = 1 +
R11r2
2Nℓ3p
dvS3 ∧ dx11,
(xi)2, x11 = R11φ, (1.3)
where µ, ν = 0, 1, · · · , 5, i, j = 6, 7, 8, 9 and 0 ≤ φ ≤ 2π . We can see this metric has an
so(4) symmetry group of rotations in the directions transverse to the M5-brane. In the
near horizon limit, the harmonic function f becomes f =
R11r2
. While, the other parts
of background (1.3) remain unchanged. Actually, if letting the radius of x11 coordinate
approach zero, then the metric (1.3) can reduce to the N coincident NS5-brane solution
in ten-dimensional string theory [5].
Here we will mainly focus on the classical dynamics of a M2-brane in the above back-
grounds (1.1) and (1.3). The dynamics of this single membrane can be described by the
Nambu-Goto and Wess-Zumino type effective action. However, for the coincident mem-
branes, unlike the coincident D-brane in string theory which can be described by the
effective action [6], their worldvolume action is still not very clear [7]. We choose the
worldvolume coordinates of membrane as x0, x1, x2, and those of M5-brane as x0, · · · , x5.
Hence M2-brane is “parallel” to the M5-brane, i.e it is extended in some of the M5-brane
worldvolume directions xµ, and point-like in the directions transverse to the M5-brane
(x6, x7, x8, x9, x11). Indeed, this configuration breaks supersymmetry completely. We can
label the worldvolume coordinates of the M2-brane by ξµ, µ = 0, 1, 2, and use reparame-
terization invariance on the worldvolume of the M2-brane to set ξµ = xµ. The position of
the M2-brane in the transverse directions, (x6, · · · , x9, x11), give rise to scalar fields on the
worldvolume of the M2-brane, (X6(ξµ), · · · , X9(ξµ), X11(ξµ)). A single M2-brane world-
volume action [8] is given by the sum of the Nambu-Goto action and the Wess-Zumino
type term in the following form
SM2 = −T2
− detP [G]µν + T2
P [A] (1.4)
where the tension of the M2-brane is expressed as T2 = 1/4π
2l3p, and P [· · ·] means the
pullback operation
P [G]µν =
GMN(X),
P [A] =
AMNL(X) . (1.5)
The indices M,N,L run over the whole eleven dimensional spacetime. And the fields
GMN , AMNL denote the metric and form field in eleven dimensions. In the following
sections, we will discuss the M2-brane classical dynamics in the above backgrounds, and
suppose that the transverse coordinates of M5-brane only depend on the time coordinate.
In this case the Wess-Zumino term in the membrane action will vanish.
2 Classical dynamics of membrane
Now let us consider the membrane dynamics in the background (1.1). Since we have
supposed that the directions transverse to the M5-brane X i are only the function of time
t, where i = 6, 7, 8, 9, 11, the pullback quantities are as following
P [G]tt = −H−
3 Ẋ iẊ i,
P [G]x1x1 = H
P [G]x2x2 = H
3 , P [A] = 0. (2.1)
after substituting the above equations (2.1) into the M2-brane action (1.4), we get
SM2 = −V T2
H−1 − Ẋ iẊ i (2.2)
where V is the space volume of the M2-brane, also i = 6, · · · , 9, 11. We can find it is very
similar to the corresponding one in [9] which is the DBI action of D-brane in the N NS5
brane background. Then through using the Legendre transformation, the Hamiltonian is
H−1 − Ẋ iẊ i
≡ V E (2.3)
where the E denotes the energy density. And the equation of motion will be
H−1 − ẊjẊj
H−1 − ẊjẊj
. (2.4)
Using this equation of motion (2.4), one can check that the Hamiltonian is conserved. To
solve the (2.4), we need the initial conditions that it is ~X(t = 0) and ~̇X(t = 0). These
two vectors define a plane in R5. By an SO(5) rotation, we can rotate this plane into
the (x6, x7) plane. Then the motion will remain in the (x6, x7) space for all time. Thus,
without loss of generality, we can study trajectories in this space.
We choose the polar coordinates
X6 = R cos θ,
X7 = R sin θ. (2.5)
Then the energy density (2.3) will become
H−1 − Ṙ2 − R2θ̇2
, (2.6)
and the angular momentum density will be
H−1 − Ṙ2 − R2θ̇2
. (2.7)
We can find this angular momentum of the M2-brane is conserved as well. From the
membrane action (2.2), we can obtain energy momentum tensor. The components of Tµν
are listed in the following
T00 = −
H−1 − Ẋ iẊ i
Tij = −T2δij
H−1 − Ẋ iẊ i, (2.8)
and the other components of stress tensor are zero. From the angular momentum Lθ
equation (2.7) and energy density E (2.6), we can get the equations of the coordinates R
and θ
Ṙ2 =
T 22 +
, (2.9)
EH(R)R2
. (2.10)
For simplicity, we can first consider Lθ = 0 case, then the radial equation is
Ṙ2 =
. (2.11)
The right hand of the above equation can’t be smaller than zero, so we get a constraint
on the coordinate R is
πNl3p
− 1. (2.12)
From the above equation, we can see if the energy density E is larger than the tension of
a M2-brane, T2, the constraint (2.12) is empty and the M2-brane can escape to infinity.
However, for E < T2, the M2-brane does not have enough energy to overcome the grav-
itational pull of the M5-brane, and then will fall down to the M5-brane from an initial
position.
Choosing the near horizon limit, hence the harmonic function becomes H = πNl3p/R
Then the equation (2.11) will be
Ṙ2 =
πNl3p
π2N2E2l6p
R6. (2.13)
Since the left hand of the equation (2.13) is nonnegative, the coordinate R has a maximal
value
πNE2l3p/T
. Also from this equation, the minimal value of R is zero. Except
for these two, there are no other extremum. But there is one inflexion between points
R = 0 and
πNE2l3p/T
. We can regard the M2-brane is at the maximal value
πNE2l3p/T
at the initial time. Due to the gravitational force of M5-brane, the M2-
brane then will roll down to the M5-brane. As the time t → ∞, the radial coordinate
R approaches to zero. We can calculate the energy momentum tensor which is Tij =
EH(R)
as the R → 0, the Tij will approach to zero. It may regard as the pressure
decreasing to zero. But we need to mention that the coordinate R can’t reach zero, since
at this point the supergravity background will be not reliable. Then the classical dynamics
of the membrane near R = 0 from the above analysis will become incorrect. Thus, in
order to use the supergravity approximation, we must constrain the coordinate R to be
larger than the planck length lp.
Now we begin to consider the nonzero case of angular momentum. From the radial
equation of motion (2.9), and after substituting the harmonic function H = 1+πNl3p/R
we can get the constraint on the radial coordinate R is
πNE2l3p
πNl3p
. (2.14)
If choosing the equal case of the above equation, the constraint will become
πNE2l3p
πNl3p
= 0. (2.15)
The above equation only has one real root which is the maximal distance that M2-brane
is separated from M5-brane.
For simplicity, we choose the near horizon limit, then the equation of motion for the
radial coordinate will become
Ṙ2 =
πNl3p
π2N2E2l6p
π2N2E2l6p
R6. (2.16)
We find that the equation (2.16) is still very difficult to solve. Instead, here, we take
some analysis for this equation. If letting Lθ = 0, then this equation will reduce to the
equation (2.13). We let the left hand of the equation (2.16) to zero, then we can get the
extremal value for R. Actually, there are two only two real extremal values of the radial
coordinate R. One is R = 0, the other is
108πNlp
3E2T2 + 12
6 + 27π2N2lp
6E4T2
− 12Lθ2
108π Nlp
3E2T2 + 12
6 + 27π2N2lp
6E4T2
. (2.17)
When Lθ = 0, the above R value will reach the
πNE2l3p/T
. As the same in the
Lθ = 0 case, between the R = 0 and (2.17) there exists a inflexion. We can suppose
that the M2-brane is at the maximal value (2.17) at the initial time, then under the
gravitational pull of M5-brane, it will monotonic approach to M5-brane. Of course for the
Lθ nonzero case, the equation (2.10) for the θ coordinate in the near horizon background is
θ̇ = Lθ
= RLθ
πNEl3p
. Thus, if the radial coordinate R reaches the value (2.17), the angular
velocity will choose the maximum, and as the R → 0, the angular velocity does also
approach to zero. The energy momentum tensor satisfies Tij = −δij
EH(R)
= −δij
πNEl3p
Thus, it again goes to zero as in the Lθ = 0 case. As mentioned in the above, near the
region R = 0, the classical background will be instability due to the strong interaction.
Hence the above supergravity analysis will become unreliable in this region.
From the first section, we already know that, after compactifying a periodic circle of
coordinate x11, the metric (1.1) will become background (1.3). In the following, we study
the membrane dynamics in this background (1.3). Here, we still suppose the directions
transverse to the M5-brane X i and X11 are only the function of time t, where i = 6, 7, 8, 9,
then the pullback quantities take the form as follows
P [G]tt = −f−
3 + f
3 Ẋ iẊ i + f
3 Ẋ11Ẋ11,
P [G]x1x1 = f
P [G]x2x2 = f
3 , P [A] = 0. (2.18)
After inserting (2.18) into the M2-brane action (1.4), we can get
SM2 = −V T2
f−1 − Ẋ iẊ i − Ẋ11Ẋ11 (2.19)
where V is the space volume of the M2-brane. This action is also very similar to action
in [9] except for the harmonic function and dimension. From the Lagrangian (2.19), we
can derive the equations of motion for the membrane in this background as followes
f−1 − ẊjẊj − R211φ̇2
f−1 − ẊjẊj −R211φ̇2
, (2.20)
R11φ̇
f−1 − ẊjẊj − R211φ̇2
 = 0. (2.21)
Due to some symmetry of this system, there are also some conserved charges. Time
translation invariance implies that the energy
H = PiẊ i + Pφφ̇− L (2.22)
is conserved. The momentum is obtained by varying the Lagrangian L,
δẊ i
T2V Ẋi
f−1 − ẊjẊj − R211φ̇2
, (2.23)
T2V R
f−1 − ẊjẊj − R211φ̇2
. (2.24)
Substituting (2.23) into (2.22), we find that the energy is given by
f−1 − Ẋ iẊ i − R211φ̇2
≡ V E. (2.25)
And since the harmonic function f = 1 +
R11r2
, then ∂if(r) = X
if ′(r)/r, and one of the
equations of motion (2.20) can be rewritten as
f−1 − ẊjẊj − R211φ̇2
X if ′
2rf 2
f−1 − ẊjẊj − R211φ̇2
, (2.26)
the other one is unchanged.
To solve these equations, we need to specify some initial conditions for the coordinates.
One condition is ~X(t = 0) and ~̇X(t = 0). These two vectors define a plane in R4. By an
SO(4) rotation symmetry, we can rotate this plane into the (x6, x7) plane. The other one
is φ(t = 0) and φ̇(t = 0). Then the motion of the membrane will remain in the (x6, x7, φ)
space for all time. Thus, without loss of generality, we can study trajectories in this space.
In addition to the energy, the angular momentum of the M2-brane is conserved as well.
It is given by
(X6P 7 −X7P 6). (2.27)
Using the expression for the momentum, (2.23), we find that
Lθ = T2
X6Ẋ7 −X7Ẋ6
f−1 − ẊjẊj −R211φ̇2
. (2.28)
Another interest quantity is the stress tensor Tµν associated with the moving M2-
brane. The component T00 denotes the energy density, so it is given by expression (2.25)
for E, with the factor of the volume stripped off. We list the components of Tµν in the
following equations
T00 = −
f−1 − Ẋ iẊ i −R211φ̇2
Tij = −T2δij
f−1 − Ẋ iẊ i − R211φ̇2,
Tφφ = −T2R211
f−1 − Ẋ iẊ i −R211φ̇2, (2.29)
and the other components of stress tensor are zero.
Due to the so(4) rotation symmetry in the transverse directions of M5-brane, it is
convenient to change to the polar coordinates
X6 = r cos θ, X7 = r sin θ. (2.30)
In these coordinates, the expressions of the energy density and angular momentum density
becomes
f−1 − ṙ2 − r2θ̇2 − R211φ̇2
, (2.31)
f−1 − ṙ2 − r2θ̇2 − R211φ̇2
, (2.32)
f−1 − ṙ2 − r2θ̇2 − R211φ̇2
. (2.33)
One can check directly that Lθ and Lφ are conserved by using the equations of motion
(2.26) and (2.21).
In order to solve the equations of motion for the given energy and angular momentum
densities E, Lθ and Lφ, we would like to solve the equation (2.32) for θ̇, and then substitute
this solution into the (2.31). Then the equation for the θ̇ is
. (2.34)
Inserting it into (2.31), (2.32) and solving for ṙ, we find
ṙ2 =
E2f 2
T 22 +
. (2.35)
Also we have the equation of φ̇
EfR211
. (2.36)
In the next, we would like to study the solutions of the equations of motion (2.34), (2.35)
and (2.36).
Firstly, we consider the angular momentum Lθ = 0 case. Then Equation (2.34) implies
that θ is constant, while the radial equation (2.35) takes the form
ṙ2 =
E2f 2
T 22 +
. (2.37)
Since the right hand side of the equation(2.37) is non-negative, then we can get the
condition 1
T 22 +
≥ 0. After substituting the harmonic function f , (1.3),
into it, we find the constraint on r (for fixed energy density E)
R11r2
− 1 (2.38)
where we can define the effective M2-brane tension is
T 2e = T
(2.39)
From the equation of constraint (2.38), obviously, if the energy density E is larger than
the effective tension of a M2-brane, Te, the constraint (2.38) is empty and the M2-brane
can escape to infinity. For E < Te, the M2-brane does not have enough energy to escape
the gravitational pull of the M5-brane, which means that it cannot exceed some maximal
distance from the M5-brane.
Under the near horizon limit, the harmonic function f will become f =
R11r2
. Then
the equation (2.38) will be
R11r2
. Thus, if r <<
, the effective tension of
membrane Te satisfies the constraint Te/E >> 1. However, r >>
, the case will
be otherwise. Indeed, in this near horizon case, we can solve for the trajectory r(t), φ(t)
exactly. Substituting the harmonic function f =
R11r2
into (2.37), we find the equation
of motion
ṙ2 =
E2N2l6p
T 22 +
r4. (2.40)
Then the solution can be obtained
L2φ +R
NR11E2l3p
t (2.41)
where we choose t = 0 to be the time at which the M2-brane reaches its maximal distance
from the M5-brane. For an observer living on M5-brane, the M2-brane reaching r = 0
will take an infinite time. Also, the M2 radial motion is similar to D-brane’s motion in
And the equation of motion (2.24) becomes
φ̇2 =
r2 − ṙ2
11 + T
. (2.42)
Substituting the solution r into equation (2.42), we can get the equation
L2φ + T
. (2.43)
Then after solving this equation, the solution can be obtained
L2φ + T
t. (2.44)
It is interesting to calculate the energy momentum tensor of the M2-brane in this
case. The energy density T00 is constant and equal to E throughout the time evolution.
However, for the parts Tij and Tφφ, we can find
Tij = −δij
Tφφ = −
R211T
. (2.45)
We see that the pressure goes smoothly to zero as r → 0, since f(r) ∼ 1/r2. But again
as the analysis in the background (1.1), this may be unreliable near the r = 0 region.
So far we have discussed the trajectories with vanishing angular momentum density
(2.32). A natural question is whether anything qualitatively new occurs for non-zero Lθ.
Just as [9], we can think as follows, the radial equation of motion (2.35) can be thought
of as describing a particle with mass m = 2, moving in one dimension r in the effective
potential
Veff(r) =
E2f 2
T 22 +
(2.46)
with zero energy. Now we discuss the properties of this effective potential Veff . In the
small r region, it will behave as
Veff(r) ≃
E2Nl3p
r2. (2.47)
For large r, the leading terms of this potential will be
Veff(r) ≃
− 1. (2.48)
If the energy density of the M2-brane is smaller than the effective tension of a M2-brane,
E < Te, then the effective potential Veff approaches to a positive constant (2.48) as r → ∞,
which means the membrane cannot escape to infinity. From the equation (2.47), we can
find that in order to have trajectories at non-zero r, the angular momentum must satisfy
the constraint
NEl3p
. (2.49)
If the constraint (2.49) is not satisfied, the only solution is r = 0. But, if the condition
(2.49) is satisfied, the trajectory of the M2-brane is qualitatively similar to that in the
Lθ = 0 case. It will approach the M5-brane and does not have stable orbits at finite r.
For the case Te >> E, the whole trajectory lies again in the region r <<
Nl3p/R11,
and one can approximate the harmonic function (1.3) by f =
R11r2
. Then the equation
(2.35) for ṙ will be
ṙ2 =
Nl3pE
E2N2l4p
T 22 +
r4 , (2.50)
with the solution
L2φ +R
R11NE2l3p − R11L2θ
NR11E2l3p − R211L2θ
NEl3p
t. (2.51)
We can find that the non-zero angular momentum can slow down the exponential decrease
of r as t → ∞. In the near horizon limit f(r) = Nl
, the solution of the equation (2.34)
for θ is
R11Lθ
ENl3p
t. (2.52)
The solution (2.51) and (2.52) mean that the M2-brane in the background (1.3) will be
spiralling towards the origin, circling around it an infinite number of times in the process.
The equation about φ is
Lφ(NE
2l3p −R11L2θ)
ENl3p(L
cosh
R11E2Nl3p − R211L2θ
ENl3p
. (2.53)
and the solution of the above equation reads
L2φ +R
NE2l3p − L2θ
R11NE2l3p − R211L2θ
ENl3p
t. (2.54)
At t = 0, the φ = 0, however, the time t → ∞, then, φ → Lφ
NE2l3p−L
Thus, the non-zero angular momentum Lθ slows down the variation of φ. From these
three solutions, we know that the M2-brane is circling along the θ direction, varying along
the φ and falling down towards the M5-brane in the process. Also, the energy momentum
tensor Tij and Tφφ will approach to zero as r → 0, since f(r) ∼ 1/r2. But we must
mention that, near the r = 0 region, the discussion may be incorrect due to the strong
coupling.
In the background (1.3), the results about the dynamics of a M2-brane have some
similar properties as studying in [9]. This can be understood that the D2-brane and
NS5-brane in IIA can be got by compactified one transverse dimension of M2-brane and
M5-brane in M theory. The solutions of equation of motion describe the M2-brane falling
towards the M5-brane. In the non-zero angular momentum Lθ, the M2-brane is spiralling
towards the M5-brane. But both in this two case, M2-brane has a angular momentum Lφ.
We need to mention that the background (1.3) is only correct in the limit of 1 ≪ r/R11.
Therefore, as the M2-brane approaches the M5-brane, the energy momentum tensor Tij
and Tφφ approaching zero may be unreliable. Since here the radial coordinate r is smaller
than the radius R11. So we are not sure whether the membrane will have the same
behavior just like the late time behavior of unstable D-brane [10, 11, 12, 13, 14].
In the above sections, we investigated the membrane classical dynamics in various
M5-brane backgrounds. There may be some generalizations, since under the Penrose
limit, the N coincident M5-brane solution (1.1) will reduce to the AdS7 × S4 geometry.
Hence one can investigate the membrane dynamics in this geometry. For the (1.1), (1.3)
and their near horizon background geometry, after calculating the classical equations of
motion of membrane from the membrane action (1.4), we can analyze the moving trajec-
tories of membrane. In some particular cases, we can get the exact solution of trajectories
of membrane. However, generally, the equations of motion is very difficult to solve. But
through analyzing these equations, we still can obtain some qualitative information about
the motion of membrane. Consequently, in the M5-brane background, the membrane will
be falling and spiralling towards to the M5-brane by the gravitational force of M5-brane.
In the near M5-brane region, i.e R (or r) being of the order of the planck length lp,
the above analysis of the classical dynamics of membrane may not be trusted, since the
method of the supergravity approximation is unreliable.
Acknowledgements
We would like to thank Yi-hong Gao for the useful suggestions and discussions.
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	Introduction
	Classical dynamics of membrane
