Challenge question: if 1kg of apples is $5 and 2kg is $10, how much is -1kg of apples?

The answer?

Priceless.

Because you could use negative-mass apples to build warp drives, travel in time, and construct a perpetual motion machine.

In fact that last one will be today’s actual challenge question.

Exotic matter – matter with negative mass - has long been the pipedream of science fiction writers, futurists, and certain rather.

.

.

optimistic researchers.

It’s the key to faster than light travel because it’s the only stuff that can curve space in the right way to hold open wormholes and construct warp fields.

And if you can travel faster than light you can also travel backwards in time.

We’ve been over those already.

And we also recently covered a very new use for negative mass: as “dark fluid”, a proposed explanation for both dark matter and dark energy.

That episode really got me thinking about the subtleties of negative mass and how it should really behave gravitationally.

Turns out it’s complicated, and to answer it we really have to question the very definition of mass.

Let’s start with mass in Newton’s physics.

It plays two roles.

First it’s the property of an object that resists acceleration.

This is encapsulated in Newton’s second law, which defines the connection between the amount of force applied and the acceleration that results.

This type of acceleration-resisting mass is called inertial mass.

Mass is also the property that both causes and responds to gravity.

It appears in Newton’s Law of Universal Gravitation.

The more massive an object, the more strongly its gravity pulls on surrounding objects and the more strongly it gets pulled by gravity.

We refer to the mass that exerts or responds to a gravitational force gravitational mass.

Clever, right?

Actually, we should split gravitational mass into active gravitational mass – that’s the mass that causes a gravitational field, and passive gravitational mass – that’s the mass that responds to a gravitational field.

These should be the same value, but it’s going to be helpful to distinguish them.

In fact both gravitational masses should be the same quantity as inertial mass.

Combine Newton’s second law with the law of gravitation and you see that although massive objects get pulled more strongly by gravity, they also accelerate more slowly.

When you calculate the acceleration of an object in a gravitational field, inertial and passive gravitational mass cancel each other out - as long as they are they same quantity.

Galileo demonstrated the equivalence of gravitational and inertial mass when he showed that objects with very different masses fall at the same rate, and that was before Newton came even along.

It’s actually pretty mysterious that the same property that defines the response to gravity also defines resistance to acceleration by all forces – including gravity.

OK, so how do negative masses work in Newtonian mechanics?

We know that the gravitational force between positive mass objects is attractive.

In fact, in Newtonian gravity, any like masses – both positive or both negative - should produce a mutually attractive force.

On the other hand opposite mass signs should repel.

That’s the opposite to electric charge, in which like charges repel and opposite charges attract.

As a side note, and at the risk of getting way too technical, this is also what quantum field theory predicts: fields with even spin have to work in the opposite way to fields with odd spin.

No time to get into why this is the case, or what the spin of a field even means, but the gravitational field is spin 2 – even - so like masses should attract and opposite should repel.

Electromagnetism is spin 1 - odd - so attraction and repulsion are flipped compared to gravity.

That seems to settle the question.

Except now we come back to the tricky relationship between inertial mass and gravitational mass.

In particular, what if passive gravitational and inertial mass is the same thing?

Newton’s second law seems to say that negative inertial masses respond oppositely to the applied force.

A repulsive force on a negative mass becomes an attractive force and vice versa.

We went over this in the dark fluid episode, but to recap: this suggests that two negative masses produce an attractive force which actually drives them apart.

Even weirder, a positive mass should attract a negative mass while at the same time being repelled by it.

A negative mass apple would still fall to the Earth, and you wouldn’t notice Earth’s infinitesimal repulsion from the apple.

But put equal positive and negative masses next to each other and they should accelerate uniformly forever.

That sounds crazy, and so we must invoke the age-old guiding principle of science, on par with Occam’s razor: the crackpot conjecture.

If something sounds too ridiculous to be true, it probably isn’t true.

It MIGHT be, but more likely you’re having a crackpot moment.

So what’s wrong with this interpretation?

In fact Newton’s law of gravitation is really only approximation of a much more complete description of gravity: Einstein’s general theory of relativity.

General relativity was, in part, inspired by the equivalence of gravitational and inertial mass.

The founding postulate of GR is the equivalence principle, which states that there’s no experiment that can distinguish between the feeling of acceleration in empty space and the feeling of weight in a gravitational field.

The equivalence principle only works if all masses experience the same acceleration in a given gravitational field, so passive gravitational mass and inertial mass have to be identical.

General relativity describes gravity as the warping of the fabric of spacetime.

The presence of active gravitational mass, and of energy, momentum, pressure, and more, change the geometry of spacetime and that new geometry defines the paths objects can travel.

A so-called geodesic path is the trajectory of an object in a gravitational field assuming no additional forces.

At first glance this tells us that any object, no matter its mass, will follow the geodesic determined by its starting position and velocity.

In fact, in pure general relativity, inertial and passive gravitational masses don’t even appear in the equations.

That should mean that a negative mass behaves the same in a gravitational field as a positive mass.

Let’s take the good-old 2-dimensional rubber sheet analogy to depict the warping of 3-D space.

Positive mass causes spacetime to curve inwards – what we call positive curvature.

In the analogy a positive mass depresses the sheet so that the trajectory of an positive mass apple bends towards the massive object.

But those trajectories only depend on the active gravitational mass of the central object, and on the velocity and starting position of the apple.

The apple’s mass doesn’t come into it.

This suggests that a positive gravitational field attracts everything, including negative masses.

So what about negative gravitational fields?

Negative mass causes negative curvature, which in our sheet analogy looks like pulling the sheet up.

Trajectories curve away from the source, and that looks like a repulsive force.

That suggests that everything, regardless of mass, should be repelled from a negative mass.

So far this seems to echo the Newtonian prediction.

It suggests that positive masses attract everything and negative masses repel everything, including each other.

A positive mass attracts and is repelled by a negative mass.

All of this assumes the simplistic case of what we call test particles – small objects moving in a much larger gravitational field.

But this should crudely translate to the case of two equal-sized objects, although not in the simple, linear style of Newtonian mechanics.

So a positive and a negative mass apple placed side by side should chase each other across the cosmos, accelerating forever.

At first glance that sounds like it breaks conservation of momentum and energy.

It doesn’t.

The positive mass apple gains positive momentum and energy as it speeds up, while the negative mass apple balances that with negative momentum and negative energy.

In fact the energies are completely unbounded – they can go to plus and minus infinity.

That’s a horribly pathological prediction and definitely still violates the crackpot conjecture.

It also violates the energy conditions of general relativity.

Now these aren’t exactly hard rules.

They are a set of conditions against negative energies that seem necessary in order for general relativity to describe a sensible universe.

In fact the mere existence of negative mass breaks certain energy conditions, but the prospect of a bottomless well of infinite negative energy breaks them all very badly, and has implications for the stability of the vacuum itself.

So what could be wrong with this idea?

In particular, with the idea of negative and positive masses accelerating each other to infinite energies?

With this sort of highly speculative science, you really have to dig down and look at the assumptions in ALL of the theory.

In this case, the basic nature of the positive versus negative gravitational fields – the way the fabric of spacetime gets stretched has to be right.

Much less clear is the way a negative mass responds to an applied force.

We assumed that passive gravitational mass and inertial mass are the same thing – and that’s required by the equivalence principle.

We then happily plugged our negative inertial mass into Newton’s 2nd law to get acceleration.

That’s highly dodgy.

But we also got the same answer thinking about the geodesics in general relativity.

Well it turns out that there were hidden assumptions even in that effort.

The geodesics of general relativity – the paths carved into the geometry of Einstein’s spacetime – are the GR analogs of Newton’s second law and give the equations of motion of particles.

They are calculated from the geodesic equation, which itself arises directly from the equivalence principle, or from Lagrangian mechanics – a more modern version of Newton’s laws of motion.

Either way, the structure of geodesics have inbuilt the assumption of the equivalence of gravitational and inertial mass.

So do we have to throw away the equivalence principle to avoid the worst craziness?

No, but I think we have the clue we need.

The real nonsense seems to be the idea of negative inertial mass.

By comparison, negative gravitational mass is kind of ok.

Negative inertial mass implies flipping the sign of the acceleration when any force is applied to exotic matter, and it also means flipping the sign of its kinetic and potential energy.

Push exotic matter and it simultaneously accelerates towards you and loses kinetic energy.

This is … incoherent to put it kindly.

It also implies that ALL fundamental forces have their directions flipped by the action of the charge of the gravitational field.

I’m pretty sure that breaks quantum field theory as well as general relativity.

So if we conclude that negative inertial mass can’t exist and we want to save the equivalence principle and the rest of physics, we need to conclude that negative mass of any type can’t exist.

So is that the answer?

Get this: I think so, but I don’t know.

WE don’t know.

More accurately, different people “know” the answer, but their answers differ.

Some say negative mass can’t exist, some say it can.

Some say that, if they do, then positive mass always attracts while negative mass always repels, while others say that like mass signs always attract and opposite mass signs always repel.

Welcome to the land of unresolved physics.

Which sounds like the perfect place for a challenge question.

Here it is: assuming that positive mass both attracts and is repelled by negative mass, come up with a way to use a pair of infinitely accelerating positive and negative mass objects to build a perpetual motion machine.

I want a device that provides a continuous energy source.

And for extra credit, based on your design, what’s the maximum power can you get out of your machine?

Let’s say the two objects are about the size of apples, but to get some decent power output make them the density of a neutron star.

Send your answers to pbsspacetime@gmail.com within 2 weeks of the release date of this episode.

Make the subject line negative mass challenge question.

Use exactly that line and check your spelling because we filter by subject line.

6 randomly selected correct-slash-creative answers will receive their choice of any piece space time swag from our brand new merchandise store.

Link to that in the description if you want to start choosing your prize.