What the Diósi–Penrose estimate shows
This calculator estimates a Diósi–Penrose gravitational decoherence timescale for a two-branch superposition in which each branch carries the same mass and the branches are separated by one effective distance. It turns the simplified self-energy gap into a decoherence time you can compare across candidate masses and separations.
That makes the tool useful for quick order-of-magnitude checks. If you are sketching a mesoscopic interference experiment, comparing levitated objects, or sanity-checking a published branch separation, it is often easier to see the mass-separation scaling first than to work through a derivation. The calculator does not claim to predict every laboratory outcome; it shows how this simplified gravitational model responds when you change the two inputs.
This is only one piece of the decoherence story. Gas collisions, black-body radiation, vibration, electromagnetic pickup, optical loss, and measurement back-action can all dominate long before any hypothetical gravitational collapse effect does. The Diósi–Penrose estimate is still useful because it provides a clean benchmark: a very short t_G suggests gravity alone would be hostile to coherence at that scale, while a long t_G says ordinary environmental noise is likely the immediate design problem.
How to choose branch mass and separation
For the gravitational decoherence estimate, the first input is the mass in one branch of the superposition. That is not automatically the full mass of the apparatus. If a levitated bead, mirror mode, membrane, or other object is being delocalized, enter the mass that actually moves between branches or the effective mass used by your model. Because m is squared in the final expression, a small mistake in mass causes a large change in t_G.
The second input is the spatial separation between the two branches. On this page it is treated as a single center-to-center distance in meters. For path-based experiments, translate the geometry into the effective branch separation rather than copying a descriptive path length unless the two are really the same. In the simplified model, increasing d raises the predicted decoherence time linearly while decreasing the gravitational self-energy gap.
Those definitions matter because the calculator is intentionally literal. If only a small internal mode is superposed, using the mass of the entire device will exaggerate the gravitational term. Likewise, if the quoted displacement in a paper is not the branch-to-branch center-of-mass separation, entering it directly can produce a neat-looking but misleading number. The output is easy to read; the harder part is choosing inputs that match the physical picture.
Diósi–Penrose formula used on this page
For equal pointlike branches, the calculator uses a compact Diósi–Penrose estimate. The first equation gives the gravitational self-energy difference and the second converts that energy into a decoherence time using reduced Planck's constant.
Those two equations are the entire engine of the calculator. There are no hidden weights or extra fitted terms: once m and d are set, ΔE_G follows directly from Gm²/d, and t_G follows from ℏ/ΔE_G. The simplicity is the point. A quick read of the algebra tells you how each input pushes the result: distance enters linearly, mass enters quadratically, and the constants are fixed SI values.
That direct structure makes the page useful for sensitivity checks. If you are exploring two nearby experimental designs, change only one variable at a time and watch the output. In this model, mass is usually the dramatic knob because doubling it shortens t_G by a factor of four, while doubling separation only stretches t_G by a factor of two. If your intuition says otherwise, the calculator is a good way to catch the mismatch before it turns into a design mistake.
Worked example: the default Diósi–Penrose inputs
With the default inputs of m = 1 × 10-14 kg and d = 1 × 10-6 m, the calculator gives a gravitational self-energy of about 6.674 × 10-33 J and a decoherence time of about 1.580 × 10-2 s. That is a concrete, easy-to-read benchmark for the simplified Diósi–Penrose estimate.
The worked example also shows how to read the result responsibly. A 15.8 millisecond gravitational timescale does not mean a real experiment will remain coherent for 15.8 milliseconds. If residual gas, thermal photons, vibrations, or control noise destroy interference sooner, those environmental channels dominate the actual experiment. The calculator is answering a narrower question: what timescale follows from the chosen branch mass and separation if you isolate the gravitational model?
Sensitivity of the Diósi–Penrose estimate to mass and separation| Scenario | Mass m (kg) | Separation d (m) | ΔEG (J) | tG (s) | Interpretation |
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| Half-mass case | 5 × 10-15 | 1 × 10-6 | 1.669 × 10-33 | 6.320 × 10-2 | Halving the mass makes the time four times longer. |
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| Baseline | 1 × 10-14 | 1 × 10-6 | 6.674 × 10-33 | 1.580 × 10-2 | This matches the default values in the form. |
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| Double-mass case | 2 × 10-14 | 1 × 10-6 | 2.670 × 10-32 | 3.950 × 10-3 | Doubling the mass cuts the time by a factor of four. |
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| Ten-times larger separation | 1 × 10-14 | 1 × 10-5 | 6.674 × 10-34 | 1.580 × 10-1 | Increasing the separation by ten stretches the time by ten. |
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The table makes the scaling easy to see. Moving from 5 × 10-15 kg to 1 × 10-14 kg cuts t_G from about 63 milliseconds to about 15.8 milliseconds, a factor of four. Doubling again to 2 × 10-14 kg cuts it by another factor of four. In contrast, keeping mass fixed and increasing separation by a factor of ten lengthens t_G by the same factor. That is exactly the d/m² pattern encoded in the formula above.
Reading the gravitational decoherence result sensibly
After you click the button, the gravitational decoherence result panel shows ΔE_G and t_G in scientific notation. A very small t_G means the simplified gravitational model predicts rapid loss of coherence. A very large t_G means gravity, in this picture, is relatively weak and may be less restrictive than ordinary environmental noise. Neither outcome is automatically good or bad; the real value is comparative.
A good check is to nudge one input and see whether the result moves in the expected direction. Increase m slightly and t_G should shrink noticeably. Increase d slightly and t_G should grow by the same percentage. If that does not happen, re-check the units. Entering micrometers as meters or grams as kilograms will overwhelm any subtle modeling assumption.
Practical gravitational scenario testing
The fastest way to use a gravitational decoherence calculator well is to treat it as a scenario explorer rather than a single-answer oracle. Start with a baseline mass and separation you believe, then vary one quantity while holding the other fixed. That approach makes the dominant scaling obvious and keeps unit mistakes from hiding 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 compact, the calculator is especially useful for back-of-the-envelope scanning. You can quickly see whether your design sits 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 model is worth the effort.
Assumptions and limitations for gravitational decoherence
This page intentionally uses the simplest equal-branch estimate for gravitational decoherence. 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 final 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 describes 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 calculator will not warn you if the chosen mass or separation is physically inconsistent with your apparatus. The page can tell you what the formula says; it cannot tell you whether the underlying experimental picture is achievable.
Common questions about gravitational decoherence
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 ΔEG = Gm2 / d, increasing d lowers the self-energy difference and therefore lengthens tG. 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.