# What is the connection between inertial/gravitational mass and relativity?

In Einstein/Infield 's The Evolution of Physics (32-35) the authors establish that inertial mass and gravitational mass are the same and then connect this to relativity.Throughout these pages I was very confused... Can someone explain the identity of inertial and gravitational mass and how it was fundamental for the theory of relativity? (Note: I have never taken a course in physics so please don't throw mathematical formulas at me- unless, of course, you can explain them and their meaning).

• Check out my answer as i think i provided what you were looking for....how einstein thought of equivalence and how it allowed him to formulate general relativity – user122066 Jul 8 '16 at 3:40
• Related: physics.stackexchange.com/q/8610/2451 and links therein. – Qmechanic Jul 12 '16 at 4:41

The gravitational mass, $m_g$, gives you the strength of the gravitational interaction while the inertial mass, $m_i$, represents the inertia of the body. The first one is the mass appearing in the Universal Gravitation Law while the second one is the mass appearing in the Newton's second law.

The equality between these masses is an empirical fact noticed since Galileo when he (allegedly) dropped different masses from the Leaning Tower and saw them hit the ground after the same time interval. There are many tests of the equality of the inertial and gravitational masses. The most precise ones make use of free fall in vacuum chambers or torsion balances. The former technique can verify the ratio $m_i/m_g=1$ up to ten digits whereas the later can verify it up to thirteen digits.

Since $m_i$ and $m_g$ are numerically equal, they cancel out in the equation of motion of a free fall body, $$m_ia=m_gg\Rightarrow a=g,$$
so any body falls with the same acceleration $g$. This result allow us to formulate the so called Equivalence Principle which can be stated as:

The motion of a gravitational test particle in a gravitational field is independent of its mass and composition. (d'Inverno)

It turns out that the Equivalence Principle is one of the key ingredients in the formulation of the General Relativity. It is due to the equality of the masses that a uniform gravitational field (such as near Earth's surface) is indistinguishable to an accelerated frame of reference in empty space (without gravitational forces). To taste this assertion just consider a simple example. Let there be someone on Earth weighting $m_gg$. On the other hand, consider this same person resting on a scale inside an upwards accelerated elevator in empty space. If we set the acceleration to be $a=g$ and apply the Newton's second law to this person we get $$F=m_i a\Rightarrow N=m_ig.$$ Since $m_i=m_g$ we see that the normal force $N$ the scale does on the body (which is the magnitude the scale reads) equals $m_gg$. By any other (local) experiment you can imagine, the person is not able to distinguish the accelerated elevator from the uniform gravitational field on Earth's surface.

• We are using the equivalence principle already in Newtonian mechanics. As soon as someone says $g=9.81m/s^2$, the equivalence principle is in full play. – CuriousOne Jul 8 '16 at 2:29

In Newtons three laws of motion, its the second law that introduces the concept of mass, here its the linking term between force applied on an object and the motion or acceleration that results. The more 'stuff' there in the object, that is the more mass it has, the harder it is to accelerate it. That is, it has more inertia; so we call this concept of mass, inertial mass.

In the first law, no notion of mass enters; recall that the first law says, that if a particle experiences no force, it is either at rest or in uniform motion; and this is why no notion of mass enters - since no force is applied. A frame in which Newtons first law holds is called an inertial frame (Einstein, himself called it a stationary frame). Its these frames that are important in Special Relativity.

Now, if we switch on gravity, this object will feel a force proportional to the amount of stuff it has; we call this gravitational mass. There is no reason that gravitational mass should equal inertial mass, but it turns out, from experiment, that they are equal; and because they are equal, mass doesn't appear in its equation of motion - the motion of a particle in a gravitational field is independent of its mass; all objects fall the same way.

Now this looks strangely like the first law, where we said that in inertial frames, the motion of an object is independent of its mass.

Einstein noted this, and extended the notion of an inertial frame to here; he said in free-fall, for example in an elevator in free-fall, Newtons first law still holds; so we should call these frames inertial (or stationary) too.

This notion was important in how Einstein conceived General Relativity; in fact, we can read GR in one way as simply altering Newtons first law, so particles at rest (ie experiencing no force), which Newton originally said simply move in straight lines at a uniform rate, now move in 'straight' lines in the curved space of spacetime, at a uniform rate given by their own clock (proper time); such lines are called geodesics.

• One should point out that Einstein didn't do anything that isn't already done in classical mechanics with regards to equivalence. The simple statement that gravity is a mass and composition independent acceleration already contains both the weak and the strong equivalence principle. As soon as someone writes Newtonian gravity without the suffixes inertial and gravitating, they have done the equivalent of what the general theory of relativity does. What classical mechanics does not do is to formulate Newtonian gravity in a Lorentz invariant way as a field theory. That is the key difference. – CuriousOne Jul 8 '16 at 19:12
• @curiousOne: actually Malament shows that one can write Newtonian gravity as a field theory. Do you happen to know if either Newton or Galileo noted the mass & composition independent acceleration? – Mozibur Ullah Jul 10 '16 at 4:13
• Unless Newton and Galileo wrote $F=F(m, \mu_i, v)$ then they probably didn't think that (they would have been wrong if they had). I have not seen a single textbook on Newtonian mechanics that teaches the above formula, but they all write $g=9.81m/s^2$ and so all textbook authors think about it and teach it as a mass, composition and velocity independent acceleration... and that's, if I am not mistake, the equivalence principle. Of course one can write gravitation as a scalar field theory, it's just wrong, which is why Einstein went for the next best thing. – CuriousOne Jul 10 '16 at 6:37
• @CuriousOne: I'm not sure I understand what you mean by your formula - I mean I understand it, but not what you mean by it; and nor what you mean 'they would have been wrong if they had' - are you suggesting had they considered the equivalence of gravitational and inertial mass then they would have been wrong; that seems, to put it briefly, bizarre. Actually, Galileo does show that in one of his dialogues that this is the case - and without any formulas being used, but he didn't put it to the same use that Einstein did. – Mozibur Ullah Jul 10 '16 at 7:02
• My point is simply that the equivalence principle doesn't hit students out of the blue sky during their first ten minutes of their first class on general relativity. We have been teaching it in full since high school and that's how Newtonian gravity is usually formulated: as an acceleration that is independent of material properties and velocity. That's correct because we don't have a single measurement since Cavendish that contradicts this, so why would we pretend otherwise? If Galileo didn't assume it, then Galileo was too cautions and... in the end, wrong. – CuriousOne Jul 10 '16 at 7:16

Einstein had his "happiest thought" about his equivalence principle one day as he imagined himself in freefall towards the earth.

A typical mind wouldve thought "okay, so i feel as though im floating" but Einstein and his fabulous mind thought about it a different way.

Einstein connected his freefall with being in motion outside of a gravitational source hurtling through space at constant acceleration. This is just Newtonian physics at this point.

Ultimately Einstein thought about the idea that any object in free fall towards any gravitational source is indistinguishable from an object in motion without gravity but at constant acceleration. This is the basis of Einsteins equivelence principle.

Imagine yourself in a windowless perfectly insulating rocket accelerating at a constant $a = 9.8 m/s^2$. In such a system there is no experiment that can determine whether you are on earth in a motionless room or in a rocket in deep space moving with constant acceleration at $9.8 m/s^2$

this is the equivelence of gravitational mass $m_gg$ and inertial mass $m_ia$. That is $m_gg =m_ia$.

There is really no good reason a priori why these two "masses" should be equivalent.

To make this more incredible einstein made the absolutely remarkable assumption that because of the equivalence principle then EVERY POINT IN SPACETIME IS LOCALLY FLAT

this is a less fancy way of saying that no matter where you are in the universe you can always find a frame of reference described by special relativity.

This is one of the main lines of reasoning that allowed einstein to write down his famous equation

$G = \kappa T$

This is a set of 16 couple equations and are in general very difficult to solve. Nevertheless physicists have been playing with Einsteins theory for 100 years and have discovered far reaching consequences from black holes to big bang cosmology.

This general relativistic field equation, arguably the greatest triumph of humanity to date.

Despite the seemingly intuitive nature of the equivalence principle scientists have been performing experiments constantly to determine if the equivalence breaks down eventually....this breakdown would hint at new physics and a paradigm shift.

However these "eotvos experiments" have continually supported the equivalence principle to quite high accuracy. Ill look up the current value of the equivalence between gravitational and inertial masses.

More on Einstein's happiest thought

• The second you call out $g=9.81m/s^2$, you are already using both strong and weak equivalence in classical mechanics, i.e. it has absolutely nothing to do with general relativity. The latter only appears on the scene if you also need a Lorentz invariant theory. – CuriousOne Jul 8 '16 at 3:58
• @CuriousOne im not sure you're familiar with the history of relativistic physics – user122066 Jul 8 '16 at 3:59
• @CuriousOne im afraid you dont understand what the OP asked. Perhaps thats the issue youre having here.......in the meantime it is good practice to read historical accounts of physics. It's utterly fascinating – user122066 Jul 8 '16 at 4:03
• It's good practice to respond to questions the way that is best suited for the person asking. Perhaps that is why your answer was down voted? Your physics isnt wrong but your approach to answering questions clearly needs some work. – user122066 Jul 8 '16 at 4:05
• Youre having a real hard time understanding this. The issue is about WHAT EINSTEIN DID TO FORMULATE GENERAL RELATIVITY. Not whether equivalence is needed for GR. Geez. – user122066 Jul 8 '16 at 4:15