What is the Principle of Equivalence and how was it determined?

[Emilie.Jung]

Equivalence principle says that gravitational forces are equivalent physically to inertial forces. Can someone explain what is meant by that and how was it concluded?

 

[atyy]

As an elevator accelerates up, you will feel yourself being pulled down. In this way, one can think of the upward acceleration of the elevator as creating a gravity that pulls you down. Similarly, when the car rounds a corner, you will feel yourself being pushed outward by a force that is like a gravitational force. These forces due to the acceleration of the elevator or car are "gravitational" in the sense that they act on everything in the elevator or the car in the "same" way.

 

[Emilie.Jung]

I was asking about gravitational and inertial forces. You didn't mention anything concerning inertial forces? I also asked about their equivalency. I would appreciate if you would explain that. @atyy

 

[atyy]

An inertial force is a force that is due to being in an accelerated frame, eg. the elevator accelerating upwards or the car rounding a corner.

 

[Dale]

Equivalence principle says that gravitational forces are equivalent physically to inertial forces. Can someone explain what is meant by that and how was it concluded?

Both are proportional to mass and neither can be detected using an accelerometer.

 

[Emilie.Jung]

Can you elaborate? @DaleSpam

 

[Dale]

Sure, what more would you like to know?

 

[Emilie.Jung]

What do you mean by neither can be detected using an accelerometer? And yes, what if they are both proportional to mass? How do those relate to my question? Thank you.

 

[Dale]

What do you mean by neither can be detected using an accelerometer?

Consider an accelerometer at rest on the ground. There is an upwards contact force and a downwards gravitational force. These two forces cancel each other, but the accelerometer reading is non zero. It detects only the upwards contact force.

Similarly, consider an accelerometer at rest in the rotating reference frame of a centrifuge. There is an inwards contact force and an outwards inertial force. These two forces cancel each other, but the accelerometer reading is non zero. It detects only the inwards contact force.

 

[loislane]

Both are proportional to mass and neither can be detected using an accelerometer.

Inertial acceleration can't be detected using an accelerometer? How else can an observer with no other external inputs know that its frame is non-inertial?

 

[Dale]

Inertial acceleration can't be detected using an accelerometer? How else can an observer with no other external inputs know that its frame is non-inertial?

By the fact that he observes motion that can only be explained by forces that are not detected by the accelerometer.

 

[loislane]

By the fact that he observes motion that can only be explained by forces that are not detected by the accelerometer.

Huh? But if its accelerometers marks zero acceleration then it is in an inertial frame by definition, why should it feel inertial forces?

 

[Dale]

Suppose you are in a non inertial lab which is accelerating uniformly in a direction that we will call "up".

Inside the lab you can throw an accelerometer and see that it travels on a parabolic path. Because of the motion you know it is acted on by a downwards pointing force. However, that force is not detected by the accelerometer as it reads 0 during the motion.

Because of the existence of this force which is detectable by the motion, but not measured by the accelerometer, you know that the lab's frame is non inertial.

 

[loislane]

So you are saying that an accelerometer attached to a noninertial lab that is uniformly accelerating doesn't detect the inertial force felt by a person on that lab? This is what we are discussing here.

 

[Dale]

So you are saying that an accelerometer attached to a noninertial lab that is uniformly accelerating doesn't detect the inertial force felt by a person on that lab?

I am saying that an accelerometer does not detect inertial forces.

The circumstances don't matter. It doesn't matter if the accelerometer is at rest or moving in the non inertial frame. It doesn't matter if the frame is uniformly accelerating or undergoing some more complicated motion.

The same is true of gravity. An accelerometer does not detect gravity either. That is one of the reasons that they are considered equivalent.

 

[A.T.]

the inertial force felt by a person

You cannot "feel" inertial forces, for the same reason accelerometers cannot detect them.

 

[loislane]

You cannot "feel" inertial forces, for the same reason accelerometers cannot detect them.

I guess we must be understanding different things by inertial force.
When riding in a bus at constant rectilinear speed I consider myself not feeling any inertial forces. When the bus becomes a noninertial frame by accelerating or taking a turn I consider the forces I feel(like being pulled back or sideways respectively ) as inertial forces,hace you never felt those?

 

[PeterDonis]

When the bus becomes a noninertial frame by accelerating or taking a turn I consider the forces I feel(like being pulled back or sideways respectively ) as inertial forces,hace you never felt those?

What you feel is the perfectly normal contact force of the bus pushing back on you, not any "inertial force" causing you to move relative to the bus.

The bus isn't really a good example because it isn't a true inertial frame; you always have the force of the floor of the bus pushing up on you. That makes it difficult to distinguish when the push changes because of the bus turning or accelerating. For a better experiment, imagine yourself floating in a spaceship in low Earth orbit, when the engines suddenly fire for an orbit change. You won't feel any force (and an accelerometer, strapped to you, won't register any reading) at the instant the engine fires, even though, at that instant, the ship becomes a non-inertial frame and you start moving relative to it due to "inertial forces". You will only feel a force when some part of the spaceship pushes on you.

 

[A.T.]

I consider the forces I feel(like being pulled back or sideways respectively ) as inertial forces,hace you never felt those?

As Peter said, the forces that you actually feel, are frame independent real contact forces and stresses in your body. That you feel them is a frame independent physical fact, so it cannot be related to frame dependent inertial forces. When you analyse the bus from an inertial frame, then there are no inertial forces anymore, but the people in the bus are still squeezed.

What you can attribute to inertial forces is the visually observed coordinate acceleration of people relative to the bus, but not the felt proper acceleration.

 

[stevendaryl]

I guess we must be understanding different things by inertial force.
When riding in a bus at constant rectilinear speed I consider myself not feeling any inertial forces. When the bus becomes a noninertial frame by accelerating or taking a turn I consider the forces I feel(like being pulled back or sideways respectively ) as inertial forces,have you never felt those?

Peter and A.T. have already addressed this, but I would like to expand on it. Suppose you are in a very large spaceship floating in deep space. You are floating above the floor of the spaceship (there's no gravity). Now, suddenly the spaceship's rockets fire, and it starts accelerating. What do you feel? You feel nothing at all. You just start moving toward the rear of the spaceship. You don't feel any forces until you hit the floor of the spaceship. At the point, what you feel is the force of the floor slamming into you---in other words, you feel contact forces between you and the floor. There is never a time when you feel any inertial forces.

 

[harrylin]

I guess we must be understanding different things by inertial force.
When riding in a bus at constant rectilinear speed I consider myself not feeling any inertial forces. When the bus becomes a noninertial frame by accelerating or taking a turn I consider the forces I feel(like being pulled back or sideways respectively ) as inertial forces,hace you never felt those?

Somewhat, but there is nothing "pulling"; the force F is caused by the accelerating bus pushing on you with mass m:
F= d(mv)/dt

Peter and A.T. have already addressed this, but I would like to expand on it. Suppose you are in a very large spaceship floating in deep space. You are floating above the floor of the spaceship (there's no gravity). Now, suddenly the spaceship's rockets fire, and it starts accelerating. What do you feel? You feel nothing at all. You just start moving toward the rear of the spaceship. You don't feel any forces until you hit the floor of the spaceship. At the point, what you feel is the force of the floor slamming into you---in other words, you feel contact forces between you and the floor. There is never a time when you feel any inertial forces.

Of course we can only feel contact forces which compress our cells; we can not identify the cause of those contact forces by feeling alone. However, we can usually identify the cause of the contact force by other means, just as loislane indicated.

 

[stevendaryl]

Of course we can only feel contact forces which compress our cells; we can not identify the cause of those contact forces by feeling alone. However, we can usually identify the cause of the contact force by other means, just as loislane indicated.

I'm not sure what you mean by "the cause of the contact force". Do you mean, what is the force, microscopically? Ultimately, I assume it's electrical forces: the electrons of your feet are being repelled by the electrons of the floor. You're certainly right, that you can't figure that out solely by what it feels like.

Or do you mean the fact that the floor is pressing up against you, yet you are still on the floor? With the hindsight of physics, we know that this is because you (and the floor) are both accelerating upward, relative to a freefall trajectory. Or do you mean the cause of the floor's acceleration upward? Yes, you certainly can't figure that out from the way it feels alone.

 

[quickAndLucky]

Check out the book "Unity of the Universe" by D.W. Sciama. He was Hawking's advisor and gives an amazing non-technical treatment of the equivalence principle and mach's principle in this book. I highly recommend it. It totally changed the way I thought about general relativity.

 

[harrylin]

I'm not sure what you mean by "the cause of the contact force". [..] or do you mean the cause of the floor's acceleration upward? Yes, you certainly can't figure that out from the way it feels alone.

When we add a descriptor such as "inertial" or "gravitational" to a following word such as "field" "or "time dilation", we commonly mean that it's the assumed physical cause of the effect. You could call the force, as you seem to suggest, acceleration force, but the descriptor inertial force is more common. As I pointed out (and as we of course all know), both acceleration and inertia are required to produce that force; that aspect of mass is often called "inertial mass".

 

[A.T.]

the descriptor inertial force is more common. As I pointed out (and as we of course all know), both acceleration and inertia are required to produce that force; that aspect of mass is often called "inertial mass".

If the frame of reference is accelerating, even mass-less objects like light undergo the same coordinate accelerations in that frame.

 

[PeroK]

I guess we must be understanding different things by inertial force.
When riding in a bus at constant rectilinear speed I consider myself not feeling any inertial forces. When the bus becomes a noninertial frame by accelerating or taking a turn I consider the forces I feel(like being pulled back or sideways respectively ) as inertial forces,hace you never felt those?

Those forces are actually an illusion of sorts. Consider sitting on a bus that acclerates. You think you are being pushed back, but actually if you try to detect any force on your front, you won't find one. But, you do feel the seat pushing on your back: it will squash your jacket, perhaps even hurt your back. So, the bus is simply pushing you forward. There is no force opposing this.

Compare this with someone pushing you against the bus seat. In this case, you will feel both forces. And, if this bus seat wasn't there and you were being pushed back, then you would feel only the force on your front.

It's the same with a car going round a bend (say turning to the right, so you end up against the left door). Even though you imagine you are being pushed to the left, you feel nothing on your right shoulder, but you do feel (and could measure) the force on your left shoulder.

 

[Dale]

@loislane there appears to be no disagreement about what is referred to by the term "inertial force". Only disagreement about whether or not you "feel" them. Human perception is extremely complicated and notoriously easy to trick, so I think that question is largely irrelevant.

The fact remains that accelerometers do not detect inertial forces. Is that much clear to you? Accelerometers also do not detect gravity. Is that also clear?

 

[loislane]

Those forces are actually an illusion of sorts. Consider sitting on a bus that acclerates. You think you are being pushed back, but actually if you try to detect any force on your front, you won't find one. But, you do feel the seat pushing on your back: it will squash your jacket, perhaps even hurt your back. So, the bus is simply pushing you forward. There is no force opposing this.

Compare this with someone pushing you against the bus seat. In this case, you will feel both forces. And, if this bus seat wasn't there and you were being pushed back, then you would feel only the force on your front.

It's the same with a car going round a bend (say turning to the right, so you end up against the left door). Even though you imagine you are being pushed to the left, you feel nothing on your right shoulder, but you do feel (and could measure) the force on your left shoulder.

I was probably not clear on what exactly it is usually referred to as inertial force by physicists. In fact I was interested in the origin of whatever comes associated with the acceleration in a noninertial frame be it in the form of motions or of contact forces like the seat against the back, obviously if one discards any contact force as inertial before hand one is left just with the motions observed as a cosequence of noninertiality of a frame.

@loislane there appears to be no disagreement about what is referred to by the term "inertial force". Only disagreement about whether or not you "feel" them. Human perception is extremely complicated and notoriously easy to trick, so I think that question is largely irrelevant.

Well. it may as well be irrelevant but on the other hand I have witnessed people attaching meaning to the fact that invariant proper acceleration is something that we can "feel" when distinguishing it of other kinds of acceleration.

The fact remains that accelerometers do not detect inertial forces. Is that much clear to you? Accelerometers also do not detect gravity. Is that also clear?

this looks mainly semantic but I would like to get clear on this, accelerometers detect 1 g on the Earth's surface, I guess it is up to you to decide if that is due to the Earth's gravitational field, or to some other more contrived cause that avoids calling it gravity, or inertial force due to the Earth's being a noninertial frame, or to reactions forces, it all looks a bit arbitrary to me.

 

[PeterDonis]

In fact I was interested in the origin of whatever comes associated with the acceleration in a noninertial frame

Which acceleration? Coordinate acceleration or proper acceleration?

The standard meaning of "inertial force" is "whatever it is that causes coordinate acceleration in a non-inertial frame". If you are in a spaceship that is in free fall, and you are floating in the center, and the spaceship's engines fire, then, relative to the spaceship, you experience a coordinate acceleration (until you hit the wall). Whatever causes that coordinate acceleration is an inertial force. And, since an accelerometer strapped to you will read zero while you are experiencing this inertial force, the inertial force cannot be detected by an accelerometer (nor will you feel it).

The force that pushes on you once you hit the spaceship wall, OTOH, is an ordinary contact force, and can be measured by an accelerometer (and you will feel it).

be it in the form of motions or of contact forces like the seat against the back

If you mix up things that are fundamentally different, it's only going to cause confusion.

if one discards any contact force as inertial

No, the contact force is not an inertial force; it's an ordinary force that can be measured in the ordinary way with an accelerometer. The inertial force is the one that can't be measured in an ordinary way with an accelerometer. See above.

I have witnessed people attaching meaning to the fact that invariant proper acceleration is something that we can "feel" when distinguishing it of other kinds of acceleration.

The reason for this is simple: proper acceleration--what is measured by an accelerometer--is a direct observable. So it must be the same regardless of our choice of coordinates.

An inertial force, however, can be made to disappear simply by changing coordinates--by choosing an inertial frame. For example, in the case of the spaceship I described above, relative to the inertial frame in which it is initially at rest, you, floating in the center of the ship when its engines fire, experience no inertial force at all; you remain at rest in the inertial frame until the spaceship's wall hits you and pushes on you. So the only force that exists at all in this frame is the ordinary contact force that can be measured in the ordinary way with an accelerometer. There's no need to talk of "inertial forces" at all, or to wonder why they can't be measured with an accelerometer like other forces can.

accelerometers detect 1 g on the Earth's surface, I guess it is up to you to decide if that is due to the Earth's gravitational field, or to some other more contrived cause that avoids calling it gravity, or inertial force due to the Earth's being a noninertial frame, or to reactions forces, it all looks a bit arbitrary to me.

The 1 g detected by the accelerometers is due to the obvious cause: the Earth pushing on them. It's an ordinary contact force measured in the ordinary way. There's no question about that.

If you drop a rock, however, and you insist on using a non-inertial frame at rest with respect to the Earth, then you need to invent a "force of gravity" to explain why the rock experiences a 1 g coordinate acceleration downward. And this "force of gravity" must be an inertial force, because it can't be measured in the ordinary way with an accelerometer (an accelerometer attached to the falling rock reads zero), and it only appears in a non-inertial frame (in an inertial frame in which the rock is at rest, there is no "force of gravity"). There's no question about that either.

The only problem comes in if you try to somehow lump the above two things together, and wonder why "inertial forces" and other kinds of forces don't work the same way, since they're both "forces". The correct response to that problem is, don't do that.

 

[A.T.]

one discards any contact force as inertial

Contact forces are frame invariant physical facts. They cannot be transformed away by a frame change, like gravity(locally) or inertial forces.

Well. it may as well be irrelevant but on the other hand I have witnessed people attaching meaning to the fact that invariant proper acceleration is something that we can "feel" when distinguishing it of other kinds of acceleration.

It is relevant whether something is frame invariant or not. "Feel" is just a synonym for "detect locally without outside reference", which is what an accelerometer does.

accelerometers detect 1 g on the Earth's surface,

1g pointing upwards, just like the normal force on the accelerometer.

 

[loislane]

So how do you call the acceleration(and associated forces) related to noninertial frames that is indeed measured by accelerometers when making the distinction with inertial frames (defined as those in which an accelerometer reads zero)?

 

[PeterDonis]

inertial frames (defined as those in which an accelerometer reads zero)?

No, that's not how "inertial frames" are defined. Inertial frames are defined as those in which an accelerometer at rest in the frame reads zero. For example, an accelerometer attached to you, floating inside a spaceship in deep space, will read zero, because you are at rest in an inertial frame. If the spaceship's engines fire, an accelerometer attached to the spaceship will no longer read zero, but that doesn't mean the spaceship disappears from the inertial frame (in which you are still at rest, with your accelerometer still reading zero).

So how do you call the acceleration(and associated forces) related to noninertial frames that is indeed measured by accelerometers

I have no idea what forces you are referring to here. But there have already been plenty of specific examples given in this thread of forces that can be measured by accelerometers, vs. "forces" that can't.

 

[stevendaryl]

this looks mainly semantic but I would like to get clear on this, accelerometers detect 1 g on the Earth's surface, I guess it is up to you to decide if that is due to the Earth's gravitational field, or to some other more contrived cause that avoids calling it gravity, or inertial force due to the Earth's being a noninertial frame, or to reactions forces, it all looks a bit arbitrary to me.

This is a very important point for you to get clear about. The accelerometer is NOT measuring the force of gravity. If you drop an accelerometer, the reading goes to zero (until it hits the ground). But the force of gravity isn't zero. So the accelerometer isn't measuring the force of gravity. That's that. It's not a matter of opinion, it doesn't depend on how you look at, it's not arbitrary.

Mathematically, we can state things this way:

Let $\vec{F}_g$ be the force of gravity. Let $\vec{F}_{up}$ be the force pushing up on the accelerometer. Let $M$ be the mass of the accelerometer.

Then the total force acting on the accelerometer is $\vec{F}_{total} = \vec{F}_g + \vec{F}_{up}$. The coordinate acceleration (relative to an Earth-centered coordinate system) is given by $\vec{a} = \vec{F}_{total}/M$. The accelerometer does NOT measure this acceleration. Instead, it only measures the non-gravitation part:

$\vec{a}_{non-grav} = \vec{F}_{up}/M$

That's what people mean when they say that an accelerometer doesn't measure gravity. It measures only the non-gravitational part of the acceleration.

Now, getting back to your claim that an accelerometer measures the acceleration due to gravity: It doesn't, in general. However, there is a special case in which it does: If an object is at REST in the gravitational field, that means:

$\vec{a} = 0 \Rightarrow \vec{F}_{total} = 0$
$\Rightarrow \vec{F}_g + \vec{F}_{up} = 0$
$\Rightarrow \vec{F}_{up} = - \vec{F}_g$
$\Rightarrow \vec{a}_{non-grav} = - \vec{F}_{g}/M$
$\Rightarrow a_{non-grav} = g$ (because $a_g = -M g$)

So it's always the case that an accelerometer measures the non-gravitational part of the acceleration. But if we choose the nongravitational part so that the total acceleration is zero, then (and only then) will an accelerometer measure the acceleration due to gravity.

Saying that it's measuring gravity is sort of like saying that you're weighing a pig when you're actually weighing a rock that happens to weigh the same as the pig.

 

[Dale]

this looks mainly semantic but I would like to get clear on this, accelerometers detect 1 g on the Earth's surface, I guess it is up to you to decide if that is due to the Earth's gravitational field, or to some other more contrived cause that avoids calling it gravity, or inertial force due to the Earth's being a noninertial frame, or to reactions forces, it all looks a bit arbitrary to me.

An accelerometer at rest on the Earth detects 1 g UPWARDS. So it definitely is not detecting gravity which is downwards. This is not a semantic difference. Physically the upwards pointing contact force is detected by the accelerometer. Physically the downwards pointing gravitational force is not detected by the accelerometer.

The only semantic component is in deciding if you even want to use the word "force" to describe gravity and inertial forces.

 

[Dale]

But if we choose the nongravitational part so that the total acceleration is zero, then (and only then) will an accelerometer measure the acceleration due to gravity.

Of course, as your math showed, you mean "measure the opposite of the acceleration due to gravity"