Gravitational Decoherence Time Calculator

What this calculator estimates

This page estimates a gravitationally induced decoherence timescale for a deliberately simple quantum superposition: two equal branches, each carrying mass m, separated by a distance d. In the Diósi–Penrose picture, the two branches correspond to slightly different spacetime mass distributions, and that difference can be assigned a gravitational self-energy gap usually written as ΔE_G. The larger that gap becomes, the harder it is for the superposition to remain coherent. The calculator turns that idea into two numbers you can inspect immediately: the estimated gravitational self-energy in joules and the associated decoherence time t_G in seconds.

That makes the tool useful for quick order-of-magnitude thinking. If you are sketching a mesoscopic interference experiment, comparing candidate masses, or checking whether a proposed separation is remotely plausible, it is often more helpful to see the scaling first than to dive into a full paper-length derivation. The output does not prove that a real device will decohere in exactly that time, but it does show how the simplified Diósi–Penrose estimate responds when you change the mass or branch spacing. In other words, it is a physics intuition builder that still produces concrete numbers.

This is not a universal answer to why a real experiment loses interference. Gas collisions, black-body radiation, vibrations, electromagnetic noise, optical loss, and measurement back-action often dominate long before any hypothetical gravitational collapse effect does. Still, the Diósi–Penrose estimate is interesting because it gives a clean benchmark. If the calculator returns an extremely short time, the model says gravity alone would already be hostile to coherence at that scale. If it returns a long time, gravity is comparatively gentle and ordinary environmental decoherence is likely the more immediate design problem.

How to interpret the inputs

The first input, mass of each branch, is the mass present in one branch of the superposition. That is not automatically the total mass of the laboratory setup. If a levitated bead, membrane mode, mirror, or other object is the degree of freedom being placed into superposition, you usually enter the mass of that delocalized object or the effective mass relevant to the model. Overcounting here is one of the most common ways to get a wildly short decoherence time. Because m appears squared, a tenfold mistake in mass changes the answer by a factor of one hundred.

The second input, spatial separation, is the distance between the two branches of the mass distribution. On this page it is treated as a single center-to-center separation in meters. For path-based experiments, that means you should translate your setup into the effective distance between the two centers of mass, not just copy a descriptive length from the apparatus unless it truly matches the branch displacement. A nanometer-scale or micrometer-scale shift can matter enormously. In this simplified model, larger separation makes the gravitational self-energy smaller and therefore makes the decoherence time longer, with a directly proportional dependence on d.

Those definitions are worth slowing down for because they determine whether the output is physically meaningful. If only a small internal mode of a device is genuinely superposed, using the full device mass will exaggerate the gravitational effect. Likewise, if the quoted path length in a paper is not the same as the center-of-mass separation, entering it directly can produce an elegant but misleading number. The calculator is intentionally transparent: it does exactly what the simplified formula says, so the quality of the result depends on the care taken with those two inputs.

Formula used on this page

For equal pointlike masses in a two-branch superposition, the page uses the compact estimate below. The first equation gives the gravitational self-energy difference, and the second rewrites the decoherence time as Plancks reduced constant divided by that energy.

Those two lines summarize the main scaling law. Double the separation while holding mass fixed and the predicted time doubles. Double the mass while holding separation fixed and the predicted time becomes four times shorter. That strong mass-squared dependence is why even modest increases in branch mass can change the answer dramatically. It is also why many toy examples with tiny masses give long times, whereas larger mesoscopic masses can produce very short times even when the separation looks small in ordinary engineering language.

If you zoom out from the specific physics, the calculator still follows the same general logic as many compact scientific tools: the result is a function of a few chosen variables, and sometimes that result can be understood as a weighted contribution of inputs. The abstract forms below are preserved here because they are a useful reminder that even specialized calculations are still structured mappings from inputs to outputs.

For this calculator, the abstract function f is especially simple. It depends on two measurable inputs, uses the constants G and ℏ, and returns numbers in SI units without hidden conversions. That simplicity makes the page good for sensitivity checks. If your goal is intuition, change only one variable at a time and watch how strongly the output reacts. In practice, the mass knob is usually the dramatic one because the dependence is quadratic, while the separation knob is linear.

Worked example

The default form values give a concrete example: m = 1 × 10^-14 kg and d = 1 × 10^-6 m. Substituting those values into the simplified estimate gives a gravitational self-energy of about 6.674 × 10^-33 J. The corresponding decoherence time is about 1.580 × 10^-2 s, or roughly 15.8 milliseconds. That is a useful baseline because it sits in a range where the number is easy to picture, yet the scaling is still dramatic when you nudge either input.

The worked example also shows why the result needs interpretation. A 15.8 millisecond gravitational timescale does not mean a real experiment will stay coherent for 15.8 milliseconds. If collisions with residual gas, thermal photons, or vibrational noise destroy interference in microseconds, then environmental decoherence dominates long before the gravitational estimate matters. The page is answering a narrower question: if you isolate the usual lab complications and apply the simplified Diósi–Penrose expression, what timescale follows from the chosen mass and separation?

The table makes the scaling transparent. Going from 5 × 10^-15 kg to 1 × 10^-14 kg cuts the time from about 63 milliseconds to about 15.8 milliseconds, which is a factor of four. Going to 2 × 10^-14 kg cuts it by another factor of four. In contrast, keeping the mass fixed and increasing the separation by a factor of ten increases the time by a factor of ten. That pattern is exactly what you should expect from t_G ∝ d / m².

Reading the result sensibly

After you click the button, the result panel shows ΔE_G and t_G in scientific notation. A very small t_G means the model predicts very rapid gravitationally induced loss of coherence. A very large t_G means gravity, in this simplified picture, is relatively weak and may be less restrictive than ordinary environmental noise. Neither outcome is automatically good or bad; it depends on what comparison you are trying to make. Often the real value of the calculator is comparative: if one design change lengthens t_G by two orders of magnitude while another barely moves it, you immediately know which knob matters more within this model.

A good sanity check is to ask whether the number moves in the right direction when you perturb one input. Increase m slightly and the time should shrink noticeably. Increase d slightly and the time should grow by the same percentage. If that is not what you see, re-check your units. Entering micrometers as plain meters or grams as kilograms will distort the answer far more than any subtle modeling assumption.

Practical scenario testing

The fastest way to use this page well is to treat it as a scenario explorer rather than a one-shot oracle. Start with a baseline value you believe, then vary one quantity while freezing the other. That approach makes the dominant scaling obvious and keeps you from hiding unit mistakes inside several simultaneous changes. A few habits help:

• Use the mass of the branch that is genuinely superposed, not the mass of everything nearby.
• Convert nanometers, micrometers, and millimeters to meters before typing them into the form.
• Compare the gravitational estimate with ordinary decoherence channels instead of reading it in isolation.
• Copy the result after each run if you are building a short list of candidate masses or separations.

Because the formula is so compact, the calculator is especially good for back-of-the-envelope scanning. You can quickly see whether you are in a regime where the gravitational hypothesis would be extremely weak, moderately relevant, or immediately severe. That is often enough to decide whether a more detailed calculation is worth the extra effort.

Assumptions and limitations

This page intentionally uses the simplest equal-mass estimate. Real Diósi–Penrose calculations can involve integrals over extended mass distributions, shape effects, density assumptions, and geometric factors that are not captured by a single separation parameter. If your object is not well approximated by two equal pointlike branches, treat the output as a rough guide rather than a finished prediction. The same caution applies when only part of an object is delocalized or when internal vibrational modes matter.

Just as important, the calculator does not include environmental decoherence. It does not model gas scattering, black-body emission or absorption, optical loss, charge noise, magnetic gradients, feedback heating, or measurement back-action. In many experimental proposals those channels dominate the practical coherence time. The result here therefore tells you about one hypothetical gravitational contribution under simplified assumptions, not the total lifetime of a real superposition in a laboratory. If you are making design or interpretation decisions, use this number alongside a broader noise budget.

Finally, note that the page assumes positive, finite inputs and outputs a direct SI result without hidden conventions. That is helpful for clarity, but it means the model will not warn you if the chosen mass or separation is physically inconsistent with your apparatus. The calculator can tell you what the formula says; it cannot tell you whether the underlying experimental picture is achievable.

Common questions

Why does mass matter so strongly? The mass appears as m² in the denominator of the time formula. That squared dependence is the dominant feature of the model. If you double the branch mass, you do not merely double the gravitational effect; you quadruple the self-energy term and quarter the predicted decoherence time. That is why careful mass interpretation matters more here than in many everyday calculators.

Does larger separation always increase the time in this page? In this specific equal-mass point-particle approximation, yes. Since ΔE_G = Gm² / d, increasing d lowers the self-energy difference and therefore lengthens t_G. In more sophisticated treatments of extended objects, the detailed geometry can matter, but this page deliberately keeps the model to the clean two-variable version that is most useful for fast comparison.

What should I conclude from an extremely short or extremely long time? An extremely short time means the simplified gravitational model would destabilize that superposition very quickly. An extremely long time means gravity alone is comparatively mild for those inputs. In either case, the next scientific question is usually comparative: is this timescale shorter or longer than the ordinary environmental decoherence time for the same experiment? That comparison tells you whether gravitational decoherence is a realistic bottleneck or merely a distant background effect.

Why are there only two inputs? Because this page is not trying to be a full finite-element gravity solver. It is a compact calculator built around the equal-mass Diósi–Penrose estimate. The advantage of that choice is transparency: you can see immediately how mass and separation compete. The cost is that you must remember the assumptions whenever you interpret the output.