[Paper Review] The KLT relations in unimodular gravity

Daniel J. Burger, George Ellis+2 more|arXiv (Cornell University)|Nov 26, 2015

Black Holes and Theoretical Physics18 references3 citations

TL;DR

This paper investigates whether the KLT relations—connecting gauge theory and gravity amplitudes—hold in unimodular gravity, a modification of general relativity that fixes the spacetime volume element. Using perturbative field theory and spinor helicity methods, the authors show that the KLT relations are preserved up to four-point tree-level amplitudes in unimodular gravity, indicating a deep structural similarity between unimodular gravity and general relativity at the quantum amplitude level.

ABSTRACT

With this article, we initiate a systematic study of some of the symmetry properties of unimodular gravity, building on much of the known structure of general relativity, and utilising the powerful technology developed in that context. In particular, we show, up to four-points and tree-level, that the KLT relations of perturbative gravity hold for tracefree or unimodular gravity.

Motivation & Objective

• To determine whether the KLT relations, which link gauge theory and gravity amplitudes in general relativity, remain valid in unimodular gravity.
• To explore the extent of symmetry and amplitude structure preservation between unimodular gravity and general relativity at the perturbative level.
• To assess whether the KLT relations survive in a modified gravity theory that decouples vacuum energy fluctuations from gravity.
• To establish a foundation for comparing quantum aspects of unimodular gravity and general relativity through scattering amplitudes.

Proposed method

• Derives the perturbative structure of the unimodular gravity Lagrangian by imposing the unimodular constraint on the metric, restricting the diffeomorphism invariance to a Weyl-transverse subgroup.
• Applies the spinor helicity formalism to compute color-ordered amplitudes for gauge theory and gravity at tree level, enabling efficient amplitude calculations.
• Uses the KLT relation structure, expressed as $ M_n = ext{KLT}(A_n, A_n) $, to compare gravity amplitudes in unimodular gravity with those in general relativity.
• Performs explicit four-point amplitude computations in unimodular gravity using vertex factors derived from the unimodular action and compares them to the standard KLT form.
• Employs momentum conservation and spinor identities (e.g., Schouten identity) to simplify expressions and eliminate reference spinor dependence.
• Verifies that the resulting four-point amplitude in unimodular gravity matches the standard KLT form, confirming the relations hold at this order.

Experimental results

Research questions

• RQ1 Do the KLT relations, which relate gravity and gauge theory amplitudes in general relativity, continue to hold in unimodular gravity?
• RQ2 How does the reduced diffeomorphism symmetry in unimodular gravity—restricted to WTDiff(M)—affect the structure of scattering amplitudes?
• RQ3 To what extent is the perturbative amplitude structure of unimodular gravity isomorphic to that of general relativity at tree level?
• RQ4 Can the KLT relations be consistently applied in a theory where the cosmological constant is an integration constant rather than a dynamical source?
• RQ5 What are the implications of KLT relation preservation for the quantum behavior of unimodular gravity compared to general relativity?

Key findings

• The KLT relations are preserved up to four-point tree-level amplitudes in unimodular gravity, indicating that the fundamental amplitude structure linking gauge theory and gravity remains intact.
• The three-point amplitude in unimodular gravity, computed using the spinor helicity formalism, matches the standard Yang-Mills three-gluon amplitude $ A_3[1^-,2^-,3^+] = rac{raket{12}^3}{\braket{23}\braket{31}} $, confirming consistency with known gauge theory results.
• The four-point amplitude in unimodular gravity reproduces the KLT form $ M_4 = -s_{12} A_4[1234] A_4[1243] $, demonstrating that the KLT kernel structure is preserved at this order.
• The absence of reference spinor dependence in the final amplitude result confirms the gauge-invariant and kinematically consistent nature of the computation.
• The preservation of KLT relations in unimodular gravity suggests that the underlying duality between gravity and gauge theory is robust under certain symmetry reductions.
• The results imply that unimodular gravity, despite its different classical symmetry structure, shares the same perturbative amplitude framework as general relativity at tree level.