Trans-Planckian Pole in Einstein gravity

Let us now consider a concrete example in which one attempts to add an extra propa-gating degree of freedom to the massless graviton hµν in the trans-Planckian region, cf.

[7]. Let the extra degree of freedom be given by a scalar graviton φ of mass m. The metric seen by an external probe-source is then given by

gµν = ηµν + hµν

M_P + ηµν

φ

M_P . (2.31)

The following discussion will show that this addition is meaningful only as long as m . MP. It becomes meaningless for m becoming trans-Planckian, i.e. form > MP.

Figure 2.3: Trans-Planckian distances are shielded by a black hole barrier. Probing poles atp²=L⁻²M_P² one has to localize energy of orderL⁻¹within the distanceL.

The corresponding black hole horizon of this energy,R_H(L)≥R_S(L) = 2L²_P/L, shields the sub-Planckian region (L < L_P) from being probed by any physical experiment.

The sub-Planckian distanceL is mapped to the macroscopic distanceR_H(L). On the right-hand-side we show a qualitative plot of the energy-distance relation. The grey

“blob” around the Planck scale indicates that at the Planck scale itself we don’t know how the precise relation between engery and distances is. Also, there is an uncertainty about the far IR black holes, i.e. for energies EIR = 2L²_P/LIR, as we cannot exclude

the possibility that at scalesLLIR gravity is modified.

However, let us first take a look at Einstein gravity. At large distances, the dynamics of the massless spin-2 graviton is described by Einstein’s equation (2.2),

G_µν = 8πG_NT_µν. (2.32)

In the weak field limit, the metric and Gµν can be expanded in powers of the dimen-sionless graviton field

gµν = ηµν + h⁽¹⁾_µν +h⁽²⁾_µν +. . . , (2.33) where we have absorbed the Planck mass from (2.31) into the definition ofhµν. We keep track of the number of gravitons interacting with one another or the source by the use of a superscript denoting the order of the interaction which is equivalent to an expansion in powers ofGN. The equations of motions (2.32) to first order are given by

h⁽¹⁾_µν = −16π G_N(T_µν^source − 1

2η_µνT^source), (2.34)

where h ≡ h^µµ, T ≡ T_α^α, and the harmonic gauge g^αβΓ^γ_αβ = 0 is employed. Indices are raised and lowered with the background Minkowski metric η_µν. To linear order, the gauge condition is given by∂^µh⁽¹⁾µν = ¹₂∂ν⁽¹⁾hand the only contribution toT_µν is coming from the energy-momentum tensor of the external source, which is taken to be a static pointlike massM withT_µν^source = δ⁰_µδ_ν⁰δ³(r)M. This gives the standard first order result for the metric perturbation,

h⁽¹⁾µν

MP

= δµν

RS

r , (2.35)

where R_S = 2G_NM is the Schwarzschild horizon of the corresponding mass M black hole of mass M. This can be compared to the full solution (2.3) and it is clear that

Figure 2.4: Gravitational field produced by a sourceT. The wiggled lines represent the emitted gravitons h_µν. At the horizon the trilinear and higher order interactions are of the same order as the one-particle exchange. Hence, to obtain any meaningful

result the series has to be resummed.

for R_S/r 1 the above solution reproduces it.⁸ Note that to this order, the signal of approaching the horizon is that h⁽¹⁾_µν becomes of order one. At the same time, by consistency, the proximity of the horizon is signalled by the second and higher order perturbations in G_N becoming of order one, i.e. the contributions from the non-linear coupling of the graviton to the source are becoming as important as the ones from the linear coupling to the source. Hence, the series has to be resummed, see also Fig. 2.4.

This signals the formation of a horizon [109, 110].

Notice, despite the corrections to the metric becoming of order one, the characteristic momenta flowing through the graviton vertices are of order 1/R_S, and thus, as long as R_S L_P, the near horizon geometry is not a probe of Planckian physics. For such sources, gravity is in a weakly-coupled, αgrav 1, albeit nonlinear regime. There is an important distinction between the nonlinear regime, characterized by the amplitude of the metric perturbation being order one, h_µν ∼ 1, which can perfectly well happen while weakly coupled in the sense αgrav 1, and the regime of strong coupling where α_grav ≥1. Entering into the nonlinear regime from the linear one simply means that the expansion of the metricg_µν in terms of h_µν breaks down and one has to work with the full metric or equivalently resum. Instead, ifαgrav ≥1, the effective field theory frame work breaks down and one has to include higher order curvature operators such as, for instance,R² in the Lagrangian since they are no longer smaller than R itself.

In order to find non-linear corrections, we have to expand (2.32) to second order in h_µν, which effectively takes into account the interaction of the graviton with its own energy-momentum tensorTµν(h). To be fully consistent one would also need to include the corrections to the energy-momentum tensor of the source Tµν⁽¹⁾. Yet, as shown in appendix A, they are negligible at this order.

The equations of motion for the graviton at second order are given by

G⁽¹⁾_µν[h⁽²⁾] +G⁽²⁾_µν[h⁽¹⁾] = 0, (2.36) where G⁽¹⁾_µν[h⁽²⁾] is the Einstein tensor expanded to first order in the metric perturba-tion evaluated for the second order perturbaperturba-tion h⁽²⁾µν. Similarly, G⁽²⁾µν[h⁽¹⁾] denotes the quadratic part of the expanded Einstein tensor evaluated for the first order metric per-turbation. We will consider the latter to be sourcing the second order perturbation

8It is furthermore instructive to remember that the Newtonian potential Φ(r) can be recovered from the metric by the identification Φ =g00−1 =h00.

h⁽²⁾µν and rewrite this contribution to the equations of motion as an graviton energy momentum tensor given by

8πGNT_µν⁽²⁾[h⁽¹⁾] = −1 2h^(1)αβ

∂µ∂νh⁽¹⁾_αβ + ∂α∂βh⁽¹⁾_µν −∂α(∂νh⁽¹⁾_µβ + ∂µh⁽¹⁾_νβ)

−1

2∂_αh⁽¹⁾_βν∂^αh⁽¹⁾_µ ^β+1

2∂_αh⁽¹⁾_βν∂^βh⁽¹⁾_µ ^α − 1

4∂_µh⁽¹⁾_αβ∂_νh^(1)αβ

−1 4η_µν

1

2∂_αh⁽¹⁾_βγ ∂^βh^(1)αγ − 3

2∂_αh⁽¹⁾_βγ∂^αh^(1)βγ)

+1

4h⁽¹⁾_µν h⁽¹⁾. (2.37)

These equations yield the standard corrections to the metric at second order in GN

[109, 110]. For instance, the zero components are given by h⁽²⁾₀₀

M_P = −1 2

R_S²

r² and h⁽¹⁾₀₀ M_P = RS

r 1 +a rRS

R_c

3!

, (2.38)

where a is a factor of order 1 and R_c is the radius of the matter distribution of the mass of the source. Taking into account the backreaction of the gravitational field of the source on itself gives a small shift in the “effective” gravitational mass of the source particle, which can be safely neglected. The corrections (2.38) are the manifestation of the fact that at the horizon, i.e. r =RS, the expansion of the metric in powers ofRS/r breaks down, and the series has to be resummed.

Let us now add the extra massive scalar graviton φfrom (2.31) to the spectrum of the theory. It does not change the gravitational field of the massless spin-2 gravitonhµν at first order, since it is only sourced by T_µν^source so that (2.35) still holds. The novelty due to the presence of the massive scalar graviton, which couples to the same static external source via g^µνTµν, is that at second order in GN, h⁽²⁾µν gets additional corrections from the coupling to the energy momentum tensor of φwhich is

T_µν^φ (φ) = ∂µφ∂νφ − 1

2ηµν(∂αφ∂^αφ + m²φ²). (2.39) These corrections are accounted for by including the contributions from (2.39) evaluated for the first oder solutionφ⁽¹⁾ = e^−mr(R_S/r) on the right hand side of (2.36). Obviously, this contribution gives only an exponentially-supressed correction toh⁽²⁾_µν.

In contrast, power-law-suppressed corrections can appear if there are couplings between φand h of the form,

φ∂ⁿh^k

M_P^n+k−3, (2.40)

where the tensorial structure is not disclosed. Such contributions can arise, for example, in non-minimally coupled gravity, and they may induce an effective source forφ,

( +m²)φ = (∂ⁿh^k)

M_P^n+k−3 + ... . (2.41)

Figure 2.5: A heavy scalar (double line) is mediating the interaction between a source T and k gravitons (wiggled lines). Integrating-out this scalar at tree-level will induce

an effective point-like interaction between the source andk gravitons.

This can give corrections to φ which are not exponentially suppressed, but only by powers of (m r)⁻¹ and (M_Pr)⁻¹. For example, evaluating the right hand side of (2.41) forh = h⁽¹⁾ and r m⁻¹ can give corrections of the order (subject to cancellations in the tensorial structure)

φ^(k) MP

∼ R^k_S r^k

1

(MPr)ⁿ⁻²(m r)² . (2.42)

The reason why these correction are not exponentially suppressed is that they arise from short range processes which do not require the propagation of virtual φ-quanta over distances larger than their Compton wavelengths. In other words, these corrections can be viewed as corrections to the metric in form of non-linear powers of exclusively massless gravitons, which appear as a result of a tree-level integrating out of a heavy scalar graviton of mass m (see also Fig. 2.5) leading to

g_µν = η_µν + hµν

M_P +η_µν (∂ⁿh^k)

M_P^n+k−3m² + ... . (2.43) To summarize, we have seen that corrections coming from a heavy gravitational degree of freedom to the Einstein metric at distances larger than its Compton wavelengths are suppressed either exponentially, or by inverse powers of its mass m and thus cannot significantly affect Einsteinian gravitational dynamics at distances r m⁻¹. For ex-ample, they cannot interfere with the formation of black holes with Schwarzschild radius R_S M_P⁻¹. This is in full accordance with the notion of decoupling of heavy states at low energies [111].⁹ Although a heavy quantum state gives negligible corrections to the metric at large distances, form . M_P, these corrections are still measurable. Thus signatures of new gravitational physics at scales m⁻¹ can, in principle, be probed at much larger scalesr m⁻¹ by precision measurements. However, ifm MP this is not true because the new degree of freedom is no longer a perturbative state. Instead, it is a macroscopic black hole which does not carry any information about the UV physics.

Take, for example, the massive scalar gravitonφof mass m introduced in (2.31). Once m M_P,φcan no longer be treated perturbatively since its Compton wavelengthm⁻¹ is smaller than its corresponding Schwarzschild radiusRS. In order to understand this, it suffices to examine the gravitational field produced by the non-relativistic particle φ by simply replacing the mass of the source M by the mass m in equation (2.34). The analysis following (2.34) immediately shows that the “particle” φdevelops a horizon of

9We rely on the decoupling theorem, i.e., the theory at low energies does not give us any information on the theory at high energies.

size Rφ = 2m/M_P² which is larger than m⁻¹. For this reason, it has to be considered as a fully legitimate classical black hole. Consequently, the perturbative analysis, in which we considered contributions of virtual φquanta, is no longer applicable. Instead, one must take into account that φ represents a black hole, and therefore any contact interaction resulting from its integrating out must be exponentially suppressed by at least an entropy factore^−S, see section 2.2.1. As a result the effective operator obtained by integrating-outφ, e.g. (2.43), can no longer feature a power-law suppressed form. In other words, by becoming trans-Planckian, φ cannot carry any information other than what is carried by a large IR black hole of the same mass. Therefore, any particle with trans-Planckian mass has to be integrated out as an ordinary classical black hole of the same mass.

We are then led to the conclusion that given the fact that any degree of freedom with massm M_P is a classical object, it becomes obvious that – no matter how sophisti-cated – there is noprocess which can probe trans-Planckian physics [7]. This includes also processes like black hole evaporation, primordial quantum fluctuation and scatter-ing experiments. Note that the non-accessibility of sub-Planckian distances is even more efficient than what we described before. In fact, as shown in [4], before the black hole for-mation an eikonal barrier may form. In this case, eikonal amplitudes which describe the exchange of many soft gravitons in terms of ladder diagrams become important prevent-ing hard energy transfers through a sprevent-ingle graviton line which would encode information about short distance physics even beforep∼M_P.

Im Dokument Non-perturbative gravity at different length scales (Seite 35-40)