# Different gravitational and inertial mass

This is going to take some imagination and assumptions (and knowledge about physics).

Imagine gravitational mass and inertial mass would be different. In classical mechanics it seems a coincidence that they are the same in our universe.

• Gravitational mass is how much objects attract other things and are attracted by gravity (passive and active gravitational mass is still the same, $m_{g1} = m_{g2}$). Newton's law of universal gravitation: $\vec{F} = G \frac{m_{g1} m_{g2}}{r^2}\hat{r}$

• Inertial mass is how much objects resist change in velocity. This is the mass from Newton's second law, $\vec{F} = m_i \vec{a}$.

Furthermore, forget anything you might know about subatomic physics or non-classical mechanics. Each atom has a gravitational mass and an inertial mass (isotopes have little effect).

Assume that somehow the universe has evolved somewhat similarly with a planet like Earth existing. Both masses are still positive.

Examples:

• Objects would not fall at the same speed even in vacuum. A half-iron, half-carbon item would rotate while falling to have the fast-falling part point down (or it might break).
• If iron would have low inertial mass and high gravitational mass, then an iron vehicle would stay on the road more firmly but would be easy to turn or accelerate.
• If copper would have high inertial mass and low gravitational mass, then a cannonball made up copper could travel spectacular distances and still have a big impact. EDIT: that is given a fixed initial velocity; more realistically it would have a fixed initial energy, in which case the trajectory is the same but the impact is higher.

I am interested in what happens on Earth at technology levels up to around 1900 (so no relativity). Not really in astronomy or in how the world came to exist under these circumstances. (Though feel free to post that for future readers if you feel so inclined). Specifically, the scenario is for a computer RPG, but no need to focus on that.

• For single materials, it's kind of like being on a planet with different gravity, perhaps that's the best way to think of it... – Mark Oct 31 '14 at 16:56

Ok, so the force of gravity is stronger or weaker on certain materials, and not linked to their inertial mass. Another way of thinking of it could be that some materials have an extra force applied, either downwards or upwards, but an upward force is never more than gravity.

Depending on how strong this contrast can get, this could make for some strange game-like structures and objects... kind of like we already see in a lot of games and CGI, where things are made impossible sizes for dramatic effect. In particular, it could make it possible for there to be very large buildings since gravity would not be so hard to resist to make a huge towering structure stable, and not as much work would be needed to assemble it in the first place. Also it would collapse more slowly, though with eventually just as much impact, so you could have your heroes dramatically escape slow-moving avalanches by running away... ;->

And you could have fairy flesh have less gravitational pull, so they could jump higher... ;->

I am interested in what happens on Earth at technology levels up to around 1900 (so no relativity). Not really in astronomy or in how the world came to exist under these circumstances. (Though feel free to post that for future readers if you feel so inclined). Specifically, the scenario is for a computer RPG, but no need to focus on that.

What would happen to technology would have to do with the specific settings for certain materials, but the main use I see is low-grav materials would be good for building large towering things (walls, bridges, tall buildings) more easily. It could help with some steampunk types of things. In particular, heavy airships and submarines could be easier to build if there are useful low-grav materials available. Low-grav materials would be handy for elevators.

Low-grav grapnel hooks would be helpful.

Low-grav projectiles would have longer range, and be more accurate at range, because the amount of "aiming high" needed to compensate for projectile drop in flight would be less compared to a higher-grav projectile at the same range.

High-grav bombs and dropping-object traps would fall surprisingly quickly. This improves accuracy particularly for longer drops by limiting the correction needed to account for initial lateral velocity, and limiting the time the target has to move out of the way before the dropped object hits.

High-grav materials could be good for securing things to the ground, and if they were round or smooth or moved by things with wheels, could be more mobile than they'd otherwise need to be. However they would mainly provide out-of-proportion downward pressure and lateral friction, although if a side force exerted more force than the lateral friction, they'd slide more easily (unless blocked by obstacles to the side). That could have some uses, though I'm not sure what except as some sort of controllable anchoring system, or maybe a hydraulic pump or bellows or for sliding trap doors or hiding passages behind sliding walls.

As has been discussed already, different objects would fall at different speeds. Why? Wikipedia led me to an interesting derivation.

Consider Newton's second law: $$F=m_ia$$ Now consider his law of universal gravitation: $$F=G\frac{m_{g1}m_{g2}}{r^2}$$ Now, because the force here is due to gravity, we set the equations equal: $$m_ia=G\frac{m_{g1}m_{g2}}{r^2}$$ But because $G\frac{m_{g2}}{r^2}=g$, $$m_ia=m_{g1}g$$ and, thus, the actual acceleration due to gravity (denoted here as $g_{actual}$) is $$g_{actual}=\frac{m_{g1}}{m_i}g$$ So, the greater the gravitational mass, or the lesser the inertial mass, the greater the acceleration.

Okay, that was boring, but I figured you might want some rationale behind the ramifications. You probably did this yourself, but someone reading this might not have, so I decided to stick it in here.

An object with larger gravitational mass would undergo a greater acceleration. Strange but true; it's an artifact of the equation. So heavy objects would fall a lot faster than lighter objects - if they had the same gravitational mass. I'll take you up on what you ended your question with:

Not really in astronomy or in how the world came to exist under these circumstances. (Though feel free to post that for future readers if you feel so inclined).

and post an answer related to astronomy, simply because I love it. I'm going to base it partly off this answer.

Planets orbit due to the force of gravity; this manifests itself as centripetal force. The relevant equation here is $$F_c=\frac{m_iv^2}{r}$$ We set that equal to $$F=G\frac{m_{g1}m_{g2}}{r^2}$$ and write this as $$\frac{m_iv^2}{r}=G\frac{m_{g1}m_{g2}}{r^2}$$ We can cancel out an $r$ and make it $$m_iv^2=G\frac{m_{g1}m_{g2}}{r}$$ We solve for velocity and find that $$v=\sqrt{G\frac{m_{1g}m_{2g}}{m_ir}}$$ Thus, the speed of planets will depend on their gravitational and inertial masses. It could also mean that Trojan asteroids might not be at stable positions (although I'm not positive about this; I could be wrong).

Given that angular velocity $\omega$ is equal to $\frac{v}{r}$, the spin of the Sun's initial protoplanetary disk could be impacted. It might be unstable, with different objects moving at different speeds. I would think that this would make the early solar system fairly chaotic; planetary formation would be a lot different. Perhaps Earth would not have formed - the original planetesimals might never have coalesced into planets.

I've obviously strayed far from the area where you wanted an answer, but I wanted to explore some of the interesting consequences when it comes to astronomy. Feel free to disregard this answer if you want to.

• Glad someone did the equations. I did them briefly on paper before posting my answer, but couldn't bring myself to type out all the mark-up. A minor note: as your equation shows, whether Jupiter or Earth orbits faster at the same radius would depend only on their individual m_g/m_i ratios, not on the absolute values of either of these. As the OP hasn't specified which elements will be "gravitationally heavy" we don't know which of Earth or Jupiter will have a higher ratio. – DeveloperInDevelopment Nov 1 '14 at 4:54
• @imsotiredicantsleep Thank you for pointing that out! I've made the change. – HDE 226868 Nov 1 '14 at 18:59
• @Vandroiy What do mean? The "actual $g$" isn't some constant, like it is for an object at a distance on Earth. It depends on the object, and to find this acceleration, we need inertial mass. – HDE 226868 Nov 1 '14 at 19:22
• @Vandroiy Ah, I see. I've never really used the Gaussian formulation (I'm only in high school, after all!). But I've seen it numerous times in partial explanations of the Einstein Field Equations. – HDE 226868 Nov 1 '14 at 20:46
• That wasn't a different perspective, it was just wrong. For the record: misunderstanding the premise, I claimed I could generate energy using two objects of which one had higher $m_g$ per $m_i$. With gravitational and inertial mass decoupled, this is equivalent to considering an electron and a proton, where $m_i$ is analog to masses and $m_g$ analog to electrical charge (which also causes attraction that scales with $r^{-2}$). You can't build a perpetuum mobile with an electron and a proton. – Vandroiy Nov 1 '14 at 23:04

The technology could go absolutely anywhere. Lists are usually off-topic, so I won't just suggest a list of possible inventions - I'm sure you can imagine them anyway. Instead, let's look at how the world might be a little different, and how that might direct the development of science and technology.

If you intend to go with a very large deviation between inertial and gravitational mass then probably your biggest problem would be earthquakes. Lots and lots of earthquakes. The reason for this is that different elements on Earth will attempt to follow different orbits around the Sun. Normally, if you combine your two equations the two $m_{2}$ terms cancel and you get an orbit that doesn't depend on the nature of the satellite. Unfortunately in your case, you have $m_{g2}$ and $m_{i2}$, which do not cancel. At the extreme end, if the largest values of this ratio differ from the smallest by a factor of thousands then the surface of the Earth will be violently ripped off into space. (At the surface of the Earth, the gravitational force due to Earth is about 1600-1700 times the gravitational force due to the Sun.) For values less than this, but still large, the surface of the Earth will only almost be ripped off into space - hence, earthquakes. Gravitational "stirring" of the Earth's core and mantle will also be exaggerated by the same ratio, increasing the heating effect due to friction. For a large enough ratio, this frictional heat generation might rival radiogenic heating (which is dominant on Earth), again leading to earthquakes as it the convection of this heat that leads to plate movements. This means that as tempting as it might be to build giant, low-tech skyscrapers out of "gravitationally light" materials, it probably isn't all that safe.

I would also expect a scientific lull leading to a delay in post-Newtonian science and technology, as deriving the equivalent of Newton's second law might be a lot harder. Attempting to derive the laws of motion by experiment will yield different results for different test materials and it would probably take some time to isolate the factor responsible (probably wasting a LOT of time on shape). We only really understand the concept of inertial mass in reference to this law, so plucking the idea from thin air in a world in which inertial mass cannot be related to any other known property (such as weight in our own) would be a truly insightful leap of genius. In your setting you can solve this issue by just having everything start a little earlier and have historical societies a little more advanced than us up to Newton and then level after.

Flight will probably be the single most significant technological deviation from our own history - with the right materials even some of Da Vinci's designs might have achieved flight. This might have profound consequences for society. The relative ease of travel afforded by flight has been credited with bringing the world together in the second half of the 20th century and for reducing nationalist and xenophobic sentiment. Your world may be a more tolerant place, if not necessarily more peaceful. Contrary to the previous paragraph, this might actually lead to faster advances in scientific development as scientists are more inclined to collaborate across borders.

For $m_{g}/m_{i}$ values that don't destroy the earth, spaceflight is probably still off the table until the development of liquid fuel rockets - Napoleonic Wars through American Civil War era solid fuelled rockets are wildly imprecise by the standards needed for spaceflight. It might be possible for a highly reactive, extremely "gravitationally light" metal (I'd go for aluminium) to always be found in nature bonded to a "gravitationally heavy" element, such that the compound isn't immediately stripped off into space. Such a strongly bonded ore could only be refined by electrolysis making it very rare in your setting, but not unheard of. By the time of your story enough material might have been gathered for the world's first attempt at a steam powered rocket launch - it would make a nice news item even if you didn't want your players going into space.

That's probably a long enough post by now. Try to keep the mass ratios within a reasonable limit (10s or at most 100s) and just use your imagination.

as you pointed out, to make this work, current sub atomic particles would mess this up. Sooo...

My first idea was that atoms are the smallest particles, but then what about the electron? Without electrons we can't have electricity (and this of course would mess up a whole host of other things! including life) So on to the next idea.

With a little hand waving certain elements are made up of different sets of subatomic particles each kind particle is affected differently by either Gravitational or inertial forces giving the element different and interesting properties we currently don't have.

Projectiles could have some very interesting behaviors and ballistics mathematics would be considerably harder. However, you could have a core that had a large inertia and a wrapper that had a low gravity, and it could allow you to fire it very accurately over a very long distance.

I was thinking building would be affected but not as much as one would think. Buildings don't move (or shouldn't) so in general it is down to tinsel strength vs. weight.

However, I could see it being very useful for anti-earthquake technology. I think it could also lead to different forms of flight much earlier, especially depending on how extreme the two properties can be. If something had an escape velocity of 200 miles an hour it could really change our modes of early travel.

• Yeah on the atomic scale it gets difficult, let's assume nuclei are elementary and electrons have low, equal masses. Electricity and chemistry should remain functional that way. – Mark Oct 31 '14 at 14:10
• I'm not sure about the anti-earthquake technology. You'd want the building to be "inertially light" for that so that they move with the surface, but "inertially light" is exactly the same thing (for the same structure) as "gravitationally heavy", which isn't ideal for building - especially for floors, roofs and ceilings. You'd probably want as "normal" a ratio of masses as possible. – DeveloperInDevelopment Oct 31 '14 at 22:10
• Actually I took it to be that inertia and gravitation are separate for each element, not necessarily that they are always opposite, and I was thinking that a very high inertial piece could be put in the basement to help absorb the earthquake tremors for the building. – bowlturner Nov 1 '14 at 12:46
• @bowlturner inertial and gravitational mass would be separate and arbitrary for each element. Lets say m_g = 7 kg, m_i = 2 kg for iron. But you can restate this as m_i = (2/7) m_g, i.e. it's "inertially light". Equally, m_g = (7/2) m_i, i.e. it's "gravitationally heavy". One implies the other. With circa 1900 technology, a solid foundation and non-Earth-destroying m_i/m_g, the inertial mass of your basement is going to be dwarfed by the momentum of the plate it sits on and will make little difference. With later levels of technology (think Tokyo), I agree there are many applications. – DeveloperInDevelopment Nov 1 '14 at 15:25
• There is no reason this can't work with the same kinds of fundamental particles and the same kinds of atoms we already have. Just assign electrons, up quarks, and down quarks different g/i ratios. That will result in neutrons and protons having different m/i ratios, and this different elements (and different isotopes of the same element) having different m/i ratios. – Logan R. Kearsley Dec 19 '17 at 3:29

This actually could work even with special relativity as there is a difference between special relativity and general relativity. Special relativity describes motion while general relativity describes gravity. Having gravitational mass different from inertial mass would still work even with the speed of light and its counter intuitive effects however it would require that gravity be something other than space time curvature. Also this could still work with quantum mechanics especially considering that the three forces that are explained by quantum mechanics are independent of inertia.

In this case gravity would be explained by particles with gravitational mass exchanging virtual gravitons between each other. How likely a particle would be to emit or absorb a virtual graviton would be proportional to the gravitational mass and the gravitational force between two objects would depend on how many virtual gravitons they would exchange between each other.

Considering that the universe does not have a net electric or color charge if inertial and gravitational mass were different then the universe might have no gravitational mass. This could be accomplished by having some objects have positive gravitational mass with others having negative gravitational mass. Because opposite gravitational masses would repel and like gravitational masses would attract there would be regions of negative gravitational mass and regions of positive gravitational mass.

Considering that rest mass is the only quantum number that is not quantized if gravitational mass was different from inertial mass gravitational mass would likely be quantized. We might imagine that quarks could have gravitational masses of -2/3, -1/3, 1/3, and 2/3, while leptons would have gravitational masses of -1, 0, and 1

Electrons that have gravitational mass would tend to accelerate thousands of times more from the same gravitational force than protons and neutrons because they would have a similar gravitational mass but a much lower inertial mass. When a star would form hydrogen atoms that have the same sign for their gravitational mass as the star would be attracted to the star while those with the opposite sign would be repelled so that stars would tend to have most of the protons and electrons of the same sign for the gravitational mass. Because elements heavier than hydrogen tend to come from fusion in stars or from supernova most atoms would have their electrons and protons of the same sign for gravitational mass.

How much a given substance would accelerate from gravity would depend mostly on the fraction of electrons. Isotopes with more neutrons would tend to accelerate less from the same gravitational force as the extra neutrons would increase the number of particles without increasing the number of electrons and so would increase the isotopes inertial mass more than the isotopes gravitational mass.

Considering that electrons could either have positive or negative inertial mass or have no gravitational mass there would be three different types of electrons. This could mean that chemistry would be much more complex as it would take more electrons to fill a shell. A shell would need an equal number of all three types of electron in order to get filled. On planets objects would be made mostly of substances that are could exist in our universe as most of the electrons would be of the same type as each other and so most atoms could only form bonds with each other that have two electrons. If a substance from a planet with negative gravitational mass was put in contact with a substance with positive gravitational mass the two substances would tend to react as they would both have a different type of electron to share and so could get their shells closer to being filled. Some substances would be mostly inert to the other substances on their own planet but would still react violently if brought in contact with a substance from a planet with the opposite gravitational mass. Many substances would be the same in every way except for their gravitational mass.

While in nature there would be different substances that would fall at different rates having a substance on a planet that falls much slower than most other substances would in general need to be produced artificially as it would need to have an unusual concentration of particles that have the opposite gravitational mass from the planet or that have no gravitational mass in order to make the substance lighter. This would mean that while anti gravity would be possible it would tend to be expensive and difficult as it would require that substances with opposite gravitational mass from the planet be extracted from the environment or mined from planets with the opposite gravitational mass.

If gravitational mass was different from inertial mass then free falling would feel different from not accelerating in empty space as different parts of your body would tend be affected by gravity differently so even in free fall you could feel which direction is down. This would also mean that you could tell if you were accelerating in zero gravity or standing still in a gravitational field and so spinning a space ship would not be sufficient to create artificial gravity especially because it would not create the feeling of free fall when someone jumps.

If inertial mass was different from gravitational mass then gravity would have no effect on space and time and so any adjustment of clocks of GPS satellites would be entirely from the relativistic effect of the orbital velocity.