Question 11.
Suppose you were given a ball with negative mass. Describe what interesting effects you might expect to see.

Question 13.
In 1974, Stephen Hawking proved that Black Holes actually radiate energy by a quantum mechanical process. The radiation obeys the Stefan-Boltzmann law given in an earlier question. The effective temperature is inversely proportional to the mass of the black hole, in fact for a black hole of mass M the temperature is
T = (M_o/M) *10 ^{- 7} K , where M_o= Mass of the Sun =2 *10 ³⁰ kg
Take the radiating surface to be a sphere with the Swarzschild radius R=2GM/c²
a) Remembering that E=mc² , argue in words that the black hole should radiate away, getting smaller and hotter as it does so.
b) Put the previous argument into maths to establish
 

c) Solve this equation to show that a black hole of solar mass would evaporate in 10⁶⁶ years.
d) Why would a solar mass black hole actually grow in our universe at the moment?
e) Black holes get hotter as they lose energy! This means that they have a negative heat capacity! Suppose you had two objects like this at about the same temperature in a perfectly reflecting container. What would happen? Is this allowed?

Answer 11
People usually start by thinking what would happen if some of this negative mass is 'dropped' . To answer this question effectively, one must realise that the word 'mass' is used in school physics for two entirely different concepts.

One idea is inertial mass and is the m that appears in F=ma. It might be better to rewrite this as a=F/m to emphasise that the acceleration is the result of a force acting on an object with inertia. The m in this equation is a measure of how difficult it is to change the motion of the object.

The second use of mass is for gravitational mass in equations like F=mg and F=GMm/r². In these equations the m is telling us how strongly the object is being affected by the gravitational pull of another object. Gravitational charge might be a clearer name.

Comparing the motion of objects in uniform gravitational and electrical fields helps explain this.

In electric fields the acceleration of different objects can be different, it depends on their charge to mass ratio. Experiment tells us that all objects fall at the same rate in a gravitational field so they all must have the same 'gravitational charge' to inertial mass ratio. School physics follows Newton in making this ratio 1 for simplicity, but the price we pay is that most people fail to realise that there are two separate concepts involved here; try asking your mechanics teacher!

So the answer to what happens when we drop our negative mass depends on if we think one or other or both of the inertial and gravitational mass is negative. If it is both then it will drop as usual, just one and it will rise. You might care to consider what pushing an object with negative inertial would do! Remember that contact forces are electrical in origin and we have no reason to reverse them.

No physicist should be happy with the idea that gravitational charge just happens to be proportional to inertial mass even though the same is not true for electric charge and inertial mass. It is obviously important to know if it is true and what the deeper reason is.

The guinea and feather experiment is, of course, crude confirmation of the equivalence of gravitational and inertial mass. Baron Roland von Eötvös (!) performed more careful experiments at the end of the nineteenth and start of the twentieth centuries which confirmed the equivalence for many different materials to within experimental error, only a few parts per billion in his case.

Einstein took this equivalence very seriously, indeed it can be seen as the foundation of his General Theory of Relativity. He claimed that gravity was just as much a fictitious force as coriolis forces (see the answer to question 9) and was a result of not doing your physics from the simplest possible point of view. For example if you are in a freely falling lift there is no gravity apparent. Newton would say that all the objects are accelerating down at the same rate, Einstein says you are looking at things from the most natural point of view so this fictitious force called gravity does not appear.

Einstein said that a gravitational field is just a complicated way of saying that we are not looking at things from a freely falling frame of reference. When we say that gravity is pulling us down against the floor we should really be saying that the floor is accelerating up against us. Two 'dropped' objects hit the floor together because it is the floor accelerating up to hit them. Einstein's equivalence principle says that you cannot tell the difference between being in an accelerating box and being in a gravitational field.

Einstein's 'simpler' view of gravity is mathematically more complicated than Newton's when you come to work out the details, indeed it took Einstein quite a few years to put his ideas in full mathematical form. However it makes predictions that differ from Newton's theory for large masses and at high speeds. Experiments, all done after Einstein's theory and most of them after his death agree with him to within experimental error and certainly contradict Newton. It is typical of physics at the highest level that one person guided by ideas of elegance or beauty or simplicity should come up with a theory that matches later evidence. Science doesn't always proceed in the way some philosophers and the National Curriculum would have us believe. You don't just look at the world and come up with theories that are some neat summary of the evidence, there is often a greater harmony that guides one's ideas.

The point of this discussion for this question is that if we believe the equivalence principle then we must take negative mass to mean negative gravitational mass and negative inertial mass. So if you drop your negative mass it will indeed fall to the Earth. Newtonian physics says that F=mg tells us that the negative mass is repelled from the Earth but a=F/m tells us that it accelerates in the opposite direction to the force and so falls. Einstein just says of course!

It is entertaining to consider to masses, +m and -m, some distance apart in an empty universe.

The diagram shows how the objects repel each other so the +m accelerates away but the -m chases after it with the same acceleration at all times. This does not violate the conservation of energy since, at all times

It should be said that as far as we know negative mass particles do not exist, antimatter has positive mass.