2.4 Trans-Planckian Gravity is IR Gravity
2.4.3 On the Weakening of Gravity at the strong-coupling Scale
increasing the momentum transfer pin a scattering experiment one can probe stronger gravitational couplings. By the time the momentum transfer reaches the scale M∗, where gravity becomes strongly coupled, black hole formation starts to take over. Any further attempt of increasing p will result in the formation of larger classical black holes. The region beyond M∗ is thus outside of the reach of physical experiments in principle. Therefore, any weakening of αgrav(p) for p M∗ has no clear physical meaning as it can never be probed. M∗ is only an upper bound on the threshold scale of black hole formation, being M∗. Therefore our proof is insensitive to the details of the theory and valid for any effective field theory of gravity with a cutoff scale M∗ <
MP. Approaching the threshold of black hole formation from the weakly coupled linear domain, the one-particle exchange is a good approximation. The scale at which it breaks down coincides with the scale of black hole formation and strong coupling. In this way, a necessary connection between the strong coupling and the threshold of black
hole formation emerges, which discloses the impossibility of probing physics at distances shorter than L∗. We shall now illustrate our general conclusion on two examples [8].
An attempt of asymptotically safe gravity in four dimensions Consider a theory where Einstein gravity is valid all the way up to the Planck scale. In this theory M∗ ≡ MP. In the deep UV regime p → ∞ the theory is modified in such a way that the gravitational coupling approaches a fixed point scaling, i.e. αgrav →α∞=const, as proposed in the Asymptotic Safety scenario [9, 10]. In order to investigate whether this behaviour could have a well-defined physical meaning, one can define an interpolating propagator of the form
∆(p) = 1 MP2p2
1 1 +α p2
∞MP2
, (2.47)
which connects the IR propagator of Einstein gravity ∆(p) = M12
Pp2 to the stipulated fixed point behaviour ∆(p) → αp∞4 forp √
α∞MP in the deep UV. In the UV limit, one finds αgrav(p) = 16πGN(p)p2 ' α∞ > 1 and α∞ is constant. In order to probe distancesr∼ √α1
∞MP, the center of mass energy needs to be of the orderE ∼√ α∞MP and the momentum transfer p∼E.
This example is similar to the scenario of an additional graviton of trans-Planckian mass m which we considered in (2.31) and below. The only difference is that now the trans-Planckian state has a negative norm. Let us ignore this sign for a moment since it does not affect our argument about the impossibility of resolving the heavy mass pole. Black hole formation cannot be influenced by the would-be asymptotically safe behavior in the deep UV since for the dynamics of the formation of a black hole of sizeRH corresponding toE∼√
α∞MP, the ghost pole is decoupled and therefore irrelevant. As a consequence, any attempt of probing the length scalesL = √
α∞−1
LP which correspond to the fixed point regime results in the formation of a black hole of macroscopic sizeRH ' 2LP√
α∞. In this case, the black hole horizon is determined by [8]
h00(RH) = 2
√α∞
MP
1 RH
h
1−α∞e−
√α∞MPRH
i
= 1, (2.48)
It is apparent that the existence of the heavy ghost pole at √
α∞MP only affects the value of the black hole horizon RH with exponentially weak corrections. Accordingly, in an attempt to probe distances smaller than the Planck length LP, a black hole with radius RH ' 2
√α∞
MP > MP−1 will be produced rendering the penetration of the trans-Planckian region impossible. Asymptotic Safety is thus rendered irrelevant before it had any chance to influence gravitational physics.
To conclude, the existence of the ghost pole, which was assumed to be responsible for the Asymptotic Safety behavior, is rendered meaningless. Moreover, the UV-IR connection of gravity indicates that it should not have been included in the first place. Indeed, as a result of the black hole barrier, any physically sensible trans-Planckian state is mapped to a macroscopic object from the IR region. However, in a consistent theory of gravity there are no negative energy classical states and the ghost pole simply cannot have any
IR counterpart. Thus it should be excluded as a conseequence of the self-consistency of the theory.
Asymptotically safe gravity with a lower cut-off scale Next, we wish to con-sider an extension to the previous example in which gravity becomes strong at a scale M5 < MP. This happens whenever new (positive norm) gravitons open up at some intermediate energies. A good example of this property is provided by five dimen-sional Kaluza-Klein theories [112, 113], in which gravity becomes higher-dimendimen-sional above the compactification scale Mc = R−1c , cf. (2.25) and below. At short distances r < Rc gravity can probe the extra dimension and becomes strong at distances of the five-dimensional Planck lengthL5 = (Rc/MP2)13.
Due to the fact that at high energies gravity can probe the extra dimension, four dimen-sional gravity becomes “weaker” at these energies. Such a behaviour could be thought of to be similar to the Asymptotic Safety fixed point scenario, but the underlying rea-son for its weakening is different. Once gravity can penetrate the extra dimension, the gravitational flux lines will also extend into this dimension. Therefore, the gravitational potential at these scales is determined by a five dimensional Gauss law which gives a gravitational potential between two probes with massesM and m
V(r)∝ M m M53
1
r2. (2.49)
Here M5 ≡ L−15 is the five dimensional Planck scale. The potential with its ∝ 1/r2 behaviour falls of faster than the usual four dimensional potential which obeys ∝1/r.
From the point of view of the four dimensional theory, the underlying five dimensional theory is imprinted into the tower of massive scalar Kaluza Klein states with masses mn =n2/R2c, where nis an integer number. These states couple universally to matter, because they are a result of the compactification of the fifth dimension. They also contribute to the scattering amplitudes of gravity and modify the propagator of these theories in four dimensions according to
∆(p) =
(MP/M5)2
X
n=1
1 p2+Rn22
c
, (2.50)
where RcM53 ≡ MP2. This means that above the scale R1
c there is a tower of massive gravitons, which makes gravity strong already at scaleM5 instead ofMP. Consequently, the shortest observable length scale in this theory isL5 ≡M5−1.
Consider now such a theory equipped with a gravitational fixed point at scalespM5. The corresponding interpolating propagator is given by
∆(p) =
(MP/M5)2
X
n=1
1 p2+Rn22
c
1 1 +α p2
∞MP2
. (2.51)
As in (2.47), there is a trans-Planckian ghost pole which mimics the fixed point be-haviour. Additionally, there exists a black hole barrier at the strong coupling scaleM5. Correspondingly, also for this case, the ghost pole cannot be probed and remains un-physical [8]. Indeed for energies required to probe the ghost pole, E ∼ √
α∞MP, the black hole horizon is macroscopicRH ' α∞Rc/M5314
M5−1, and the associated state belongs to the classical gravity region.
We have seen that also in this example Asymptotic Safety has no physical meaning.
The black hole barrier, which maps the trans-Planckian region to classical IR gravity, precludes probing distances where the fixed point behaviour might become relevant.