In a gravitational field, all objects fall at the same rate if other forces, like air resistance, can be ignored. The larger masses have a greater attraction to the source of the gravitational field (the planet, in our case) but they also have a greater inertia, which EXACTLY compensates for the greater gravitational attraction.

Einstein said that inertial forces were the same as gravitational forces. The most common description is that, if one were in a sealed room on Earth, they would feel gravity pulling them down. However, if they were in a room in outer space, but with a rocket accelerating them at a gravity, it would be exactly the same.

Einstein's conslusion was that intertial forces weren't "like" gravitational forces, they ARE gravitational forces. From the fact that greater inertia counteracts greater gravitational attraction exactly, there is good reason to believe they're the same thing. However, they seems like different things to me. If someone has a different perspective that can show why gravity and inertia are the same, please let me know

"Greater inertia counteracts greater gravitational attraction exactly": there's nothing coincidental about that, as far as I can see.

I don't really get this reasoning. The forces on an object must balance. If the forces don't balance then wouldn't the object have infinite acceleration, which would then give the object infinite energy? Nature can't allow that. The Universe would blow up, or something.

Gravity does work on an object by turning gravitational potential energy into kinetic energy. An imbalance of forces is impossible here. Finite gravitational potential energy can't create infinite kinetic energy. But since Einstein was an expert in thinking outside the box he must have had a much deeper reasoning why inertial forces and gravitational forces are equal. The proof is in Einstein's results. He got so much correct.

I don't understand the trampoline analogy either. How does the trampoline relate to space in the real world. Is the direction of space defined as the path that light takes through space? How does any of that relate to the trampoline and gravity?

goes back to Galileo and the leaning tower of pizza. It was believed that heavier object fell faster. It's certainly true that the attractive force is greater - pick up a feather, then pick up a boulder and this will become immediately apparent. The attractive force is proportional to the mass of the object, penny or boulder. However, the greater inertial of the boulder will cause it to fall at the same rate as the penny if they are dropped.

Now, instead of the attractive force being caused by gravity, imagine that the Earth, penny, and boulder are electrically charged. Say the Earth in negatively charged (probably Fred Phelp's fault), and the penny and boulder have equal positive charged. They will be attracted to the Earth with equal forces, due to the equal charges. However, their rates of "falling" to the Earth will be very different. This is due to the boulder having a much higher inertia.

so the question becomes, why are gravitational forces linked to inertial forces, while electric (electromagnetic, strong, or weak) forces are not linked to inertia? Mass appears to have a special connection to gravity and inertia that the other forces don't have.

BTW, the boulder and penny have different forces - an imbalance, as you put it earlier. Inertia compensates for it. Imbalances won't cause infinite acceleration, they cause non-zero acceleration.

a physicist determines all of the eternal forces that are applied to the object and any unbalanced external force that is applied causes the object to accelerate in that direction, using the simple equation: F=MA (force = mass of the object times acceleration). So if the physicist knows the mass of the object and the unbalanced external force he can solve for the acceleration of the object.

In the total equation, the countering force from the inertia of the object balances the equation. The equation must balance. If it didn't balance, something terrible must happen. The countering force from the inertia of the object must exactly equal the external force or the object would continue to go faster. If it didn't ever balance, the object would go infinitely fast, gather infinite energy, and the Universe would explode. That would suck.

An object only being pulled by gravity is accelerated by one G. So F=MG. That's the external force. As with any equation, the internal inertia of the object must balance the external force. This equation is simple, partly because both the force of gravity and inertia are proportional to mass. But that doesn't necessarily mean that gravitational and inertial mass are equal. Einstein says they are. I presume he is correct, but he must have a deeper reasoning than just the fact that the external force of gravity and the countering force from the inertial mass must balance.

Since electrical charge isn't proportional to mass, different mass objects with equal charges will fall at different accelerations. That does make the calculations a little more complicated.

It makes sense that a bolder accelerates at the same rate as a feather in a gravitational field since you can think of the bolder as a dense collection of feathers. Why would the feathers accelerate faster if combined?

When I took dynamics (mechanical engineering, study of objects in motion), inertia developed the "force" that made all forces sum to zero, it just required a certain acceleration. And I love your description of a boulder as a dense collection of feathers

There have been quite a few experiments done to determine why gravitational and inertial mass are equal. I remember reading something in the last year or so about carefully dropping to objects or highly different masses in vacuum and measuring their accelerations. They came out equal within the accuracy of the experimental apparatus. When two things are equal with such high precision, it can't be just a coincidence, which is why physicists believe there has to be a deeper connection. Unfortunately, that will likely have to wait for a quantum gravitational theory.

"Inertia" is something measured in kg - it's the same thing as mass. And mass is not force. Inertia is something that lets you relate net force to acceleration. And when forces do not balance, the result is not infinite acceleration, but acceleration inversely proportional to inertia.

The trampoline analogy only makes sense if you understand the trampoline as analogous to spacetime. A common illustration of Einstein's ideas is to imagine a bowling ball on a trampoline and a smaller ball "orbiting" the bowling ball. The bowling ball is like the Sun, the small ball Earth, and the interaction occurs through the intermediary of the warped sheet of the trampoline. "Matter tells space how to curve, and space tells matter how to move" is the slogan version of this.

Now the blog is trying to explain gravitational waves... those imagine that this trampoline sheet also supports vibrations - those would be the gravitational waves.

Also... I found a good brief introduction to Einstein's equivalence principle. The whole view of gravity inspired by Einstein is that it's not a "force" the way, say, electromagnetism is a force, but rather a basically geometric quantity, locally indistinguishable in its dynamic effects from acceleration. I guess in aviation they run the analogy in reverse - "pulling Gs" is just accelerating, and it turns out the acceleration of a body in freefall near Earth's surface is a handy unit for measuring those accelerations!

Einstein said that locally, the effects of gravity and acceleration were identical - that they were equivalent. You may be confusing the language with the invariant spacetime interval between events of special relativity.

Why inertial mass and gravitational mass are the same really is a deep mystery. I don't think inertia is a force at all, however, so it's hard to react to what you say about "inertial forces." Perhaps you mean what I've heard referred to as "pseudo-forces" like centrifugal forces, Coriolis forces, etc. whose hallmarks are that they are proportional to mass and arise in accelerating reference frames. The similarity between these forces that arise only in accelerated frames and gravity helps inspire the notion that there may be some deep sense in which gravity might be regarded as a "pseudo" force in close analogy to all the other forces that disappear when you analyze motion in an inertial reference frame.