Time-of-Flight Mass Analysis Under a Lowered Speed of Light: Relativistic Limits on Resolving Power and Detection for the Analytical Laboratories of the Black Domain Plan

1. Introduction

The technical review of the Black Domain Plan treated the analytical laboratory as a solved problem. Its working assumption, stated once and never revisited, was that instruments inside a black domain would continue to function with their clocks and flight times rescaled to the lowered speed of light, so that a laboratory's capacity would survive the transition intact if slower (PDC Declassified Holdings, 3 BrE). That assumption is wrong for the time-of-flight mass spectrometer, which was the most common analytical instrument in the review's own inventory, and it is wrong for reasons that the review had every means to see.

After the coordinate broadcast, three survival strategies were weighed, and the Black Domain Plan was one of them. A black domain lowers the vacuum speed of light within a region of space to 16.7 km s^-1, slow enough that nothing inside can leave and the region announces itself to the cosmos as harmless. The Plan was set aside in favour of the Bunker Project, and the review that costed it is now a historical document. It is still worth getting right. Its estimates of laboratory continuity have been cited since in Bunker-Era planning for low-velocity environments, and the reasoning error behind them recurs wherever an instrument is assumed to scale with a single constant.

Here we model a conventional TOF analyser under the lowered light speed, validate the model against archived relativistic data, and work through the failure modes that the rescaling assumption hides.

2. System Description

The reference instrument is the linear TOF analyser most often listed in the review's inventory. Ions are accelerated through a potential of 20 kV, drift through a 1.0 m field-free region, and strike a detector whose arrival times are recorded against a start pulse with 1 ns precision. We also model the same analyser at 5 kV, 500 V, and 100 V, and for ions at m/z 1,000, 10,000, and 50,000, which spans peptides, intact proteins, and the megadalton assemblies that our laboratory works on. Energy spread, turn-around time, and space charge are left out, so the resolving powers reported here are upper bounds set by kinematics and timing alone.

Our model is deliberately narrow. Following the review, we take the black domain to lower the limiting speed for the free motion of ions and for signal propagation, while leaving atomic and molecular structure intact, since otherwise nobody could live inside one. We do not model what the lowered speed does to bound electrons, to chemistry, or to the electronics, beyond the propagation delay of signals in cables.

3. Analysis and Model

The controlling quantity is the ratio of an ion's kinetic energy to its rest energy. Under the conventional speed of light that ratio is negligible for any ion in any laboratory analyser. Under a limiting speed of 16.7 km s^-1 it is not. The rest energy per charge falls to 2.89 keV at m/z 1,000, 28.9 keV at m/z 10,000, and 145 keV at m/z 50,000, so a 20 kV acceleration gives a Lorentz factor of 7.92 for the smallest ion, 1.69 for the middle one, and 1.14 for the largest. The m/z 1,000 ion leaves the source at 99.2% of the limiting speed.

Resolving power in a TOF analyser depends on how strongly flight time changes with mass. Near the speed limit, velocity barely changes with mass at all, since every light ion is moving at almost the same speed, and flight times crowd together. We computed flight time from the relativistic velocity at each mass and took resolving power as the mass divided by the mass interval that one nanosecond of timing error spans, and Table 1 sets out the results for each mass and voltage.

The penalty falls hardest on small ions. At 20 kV and m/z 1,000, resolving power drops from 8,050 to 855, an order-of-magnitude loss, and the flight time stretches from 16.1 to 60.4 µs. At m/z 10,000 it falls from 25,500 to 16,300, and at m/z 50,000 from 56,800 to 51,500, a loss of 9%. A laboratory planning for this environment would find its peptide work collapse first and its native work on large assemblies survive almost unchanged, which inverts the usual ranking of what is hard in mass spectrometry.

Staying out of the relativistic regime is the remedy. At 500 V, the Lorentz factor for m/z 1,000 is 1.17, and resolving power recovers to 44,900 against 50,800 conventionally, or 89%. At m/z 10,000 it is 159,000 against 166,000, or 95%. The cost is duty cycle and transmission, since slower ions take longer to cross the analyser and are harder to keep focused, but those are costs that conventional ion optics already knows how to pay.

Ion mobility is untouched by any of this. Drift-tube ions move at tens of metres per second, some three orders of magnitude below 16.7 km s^-1, and their behaviour is set by collisions with the buffer gas, which damp external effects on the ion very strongly (Rauch-Ibáñez, 11 BkE).

4. Validation Against Field Data

No black domain has ever been made, so there is no field instrument to test against. What the model needs is a check on its kinematics, and that can be had from any system where kinetic energy is a substantial fraction of rest energy. Electrons supply one. An electron's rest energy is 511 keV, and an m/z 10,000 ion at 5 kV under the lowered light speed has the same energy-to-rest-energy ratio as an electron accelerated through 88 keV. The pre-Crisis and Crisis-Era record holds 214 archived flight-time measurements from relativistic electron spectrometers between 50 and 300 keV, collected and re-digitised by the Registry (Fleet International Academies, 8 BkE). Run with the electron mass and the conventional speed of light, our model reproduces those flight times with a mean absolute error of 0.08% and no trend with energy.

The kinematics are sound, and the resolving-power penalty in Table 1 follows from them directly.

5. Failure Modes

Two failures lie outside the resolving-power calculation, and the first of them is the more serious. Every conventional TOF detector converts an ion strike into free electrons and multiplies them through a cascade, in a microchannel plate or a discrete-dynode multiplier. The rest energy of a free electron under a limiting speed of 16.7 km s^-1 is about 1.6 meV, smaller than thermal energy at room temperature. Under the review's assumption that atoms and molecules survive inside the domain, the lowered limit cannot apply in the ordinary way to bound electrons, and the review offers no account of where free electrons stand. A detector that depends on a secondary-electron cascade therefore has no defined physical basis inside a black domain, and no design should rely on one. Image-current detection, which reads charge as the current induced on a pickup electrode and has no free-electron stage, is the scheme that avoids the question altogether, and it already reaches integer charge precision for single ions above 10 MDa (Rauch-Ibáñez, 11 BkE).

The second failure is timing. Signals in a cable travel at about two thirds of the limiting speed, roughly 11 km s^-1, so one metre of cable adds some 90 µs of delay, longer than the entire flight time at m/z 1,000 under the conventional speed of light. Start pulses, detectors, and digitisers must sit within centimetres of one another, and every delay must be calibrated in place, since the instrument's timing reference is no longer separable from its geometry.

6. Conclusion

So the review's working assumption fails. A time-of-flight mass spectrometer inside a black domain does not simply run slower. Its resolving power collapses for small ions at ordinary voltages, its detector rests on free-electron physics the review left undefined, and its timing depends on the length of every cable. The laboratory-continuity figures in the review counted instruments that would not have worked as specified, and any Bunker-Era planning document that carried those figures forward carries the same error.

A workable analytical laboratory under the Plan would have run its analysers below 1 kV, detected ions by image current, and kept its timing electronics at the flight tube. That is the specification the review should have costed. In the Lurker framework that our laboratory works within, the Luo-Ji / Ye axioms hold only locally, the Lurker (also 'Spirit', 'Spirit Mother' in some translations) is the deeper author of cosmic history, and a black domain is one of many safety notices the cosmogonic record contains. None of the kinematics here depends on that reading. They hold for any instrument, built by anyone, under a lowered speed of light.