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Equation 7 · Part 2 · How Much of Gravitationally Induced Entanglement Is in the Eye of the Frame

Both masses on L

φent=φLL+φRR−φLR−φRL\varphi_{\text{ent}} = \varphi_{LL} + \varphi_{RR} - \varphi_{LR} - \varphi_{RL}
φLL\varphi_{LL}

What this part means

The gravitational phase when A takes its L branch and B takes its L branch. It enters with a plus sign in the invariant combination.

Its job in the formula

The LL phase is added to the RR phase so matching-path contributions can be compared with crossed-path contributions.

The passage around this formula

None of this makes the experiment ambiguous, and this is the paper’s second and sharper result. The branch separations that enter the gravitational phase each mass’s superposition picks up are relational quantities — distances between mass A’s branches and mass B’s branches — and relational quantities do not change when you change which system you have designated the reference frame. Write φij\varphi_{ij} = G mAm_A mBm_B t / (ℏ\hbar dijd_{ij}) for the phase a pair of branches i,j ∈\in \{L,R\} accumulates over interaction time t at separation dijd_{ij} , with G Newton’s constant and ℏ\hbar the reduced Planck constant. Three of the four possible combinations of these branch-pair phases can always be removed…

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Learn the underlying idea

A quantum phase is an angle attached to a complex amplitude. One path’s phase can be changed by convention; phase differences between alternative paths can change interference and entanglement.

Open the illustrated quantum phase and two-path entanglement guide →

See this notation across published equations →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.

Further reading for this equation