# B Measuring Fictitious Forces in a Closed Box

#### metastable

I was reading the following article:
https://en.wikipedia.org/wiki/Fictitious_force

When I came across this passage:

"This led Albert Einstein to wonder whether gravity was a fictitious force as well. He noted that a freefalling observer in a closed box would not be able to detect the force of gravity; hence, freefalling reference frames are equivalent to an inertial reference frame (the equivalence principle)"

So suppose I am in a free falling closed box and there is no source of thrust. I have a tool to measure centrifugal forces consisting of 3 springs (X,Y,Z), each with 2 weights at each end M1 & M2. I perform a measurement of each spring and determine that one of the springs is stretched. From this single measurement of the state of the spring can I determine with certainty whether fictitious forces are present? How can I differentiate between these 2 scenarios: A) the box I am in is rotating or B) I am near an event horizon and the gravitational gradient between masses M1 and M2 is sufficient to stretch the spring

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#### Dale

Mentor
A) the box I am in is rotating or B) I am near an event horizon and the gravitational gradient between masses M1 and M2 is sufficient to stretch the spring
The equivalence principle only applies when the size of the box is small enough that gravitational gradients are undetectable.

What you want to use is a 6 degree of freedom accelerometer. Not what you have described. I don’t think that the configuration you described is possible.

#### A.T.

I have a tool to measure centrifugal forces consisting of 3 springs (X,Y,Z), each with 2 weights at each end M1 & M2. I perform a measurement of each spring and determine that one of the springs is stretched.
If only one of the orthogonal springs is streched, then it's obviously not a centrifugal force.

#### metastable

If only one of the orthogonal springs is streched, then it's obviously not a centrifugal force.
Yes, that seems to be a potential way to differentiate between centrifugal and gravitational forces.

He noted that a freefalling observer in a closed box would not be able to detect the force of gravity
^So are we saying it is possible to detect the force of gravity as a free falling observer?

• Dale

#### Ibix

So are we saying it is possible to detect the force of gravity as a free falling observer?
No. We are saying that - with sufficiently precise measurements - it is possible to detect tidal gravity. This doesn't give you the magnitude of what is typically referred to as "the force of gravity".

Depending on the details of the gravitational field, you may or may not be able to distinguish this from other fictitious forces.

#### Dale

Mentor
So are we saying it is possible to detect the force of gravity as a free falling observer?
No. Did you read my comment?

#### PeterDonis

Mentor
suppose I am in a free falling closed box and there is no source of thrust. I have a tool to measure centrifugal forces
If you're in a free falling closed box, where are centrifugal forces going to come from?

#### PeterDonis

Mentor
the box I am in is rotating
If the box is rotating, it's not freely falling. The box's center of mass might be, but the box as a whole is not.

• vanhees71

#### pervect

Staff Emeritus
The simplest form of the equivalence principle, the "Weak" equivalence principle, just says that all objects fall at the same rate due to gravity. So if you drop various test masses, made of different materials, from the same spot, you'll find they all follow the same path.

Note that if gravitational mass and inertial mass (in the Newtonian paradigm) were different, different materials would fall at different rates. So this form of the equivalence principle is basically an experimental way of saying that inertial mass and gravitational mass, in GR, are the same, there is really just one "mass", not two.

You can enclolse two test masses in a box, and the box and the test masses. Because the Earth is mostly spherical (it's not quite, so this is an approximation), the box, and it's contents, will fall towards the center of the Earth with identical accelerations, to a good approximation.

With a large enough box, though, one will see various effects that can be ascribed to tidal forces. Two balls dropped from different heights will have different accelerations as gravity (in the Newtonian sense) gets weaker with altitude, so the distance between the balls will increase as they and the box they are in fall.

Two balls dropped from the same height will both travel twoards the center of the Earth, so their paths will converge, and the distance between the balls will decrease. This is another aspect of tidal force. Tidal forces can either "stretch" or "compress". This can be seen in Newton's theory.

In GR, we can equate these Newtonian tidal forces to something else, but there's not a lot of point in going through this if one isn't sufficiently familiar with tidal forces in the first place.

Understanding the behavior of test masses in a falling box in terms of Newtonian gravity in detail is one big step towards understanding tidal forces.

Understanding tidal forces is a good thing, and becomes very useful in understanding GR. But as far as understanding the equivalence principle goes, the focus is on the fact that different materials dropped under identical conditions, fall at the same rate. Making the conditions identical is part of setting up the experiment.

#### A.T.

Yes, that seems to be a potential way to differentiate between centrifugal and gravitational forces.
It's not a potential way, but pretty straightforward.
So are we saying it is possible to detect the force of gravity as a free falling observer?
You can differentiate it from a centrifugal force, but locally you cannot tell it from a linear inertial force.

#### metastable

From this single measurement of the state of the spring can I determine with certainty whether fictitious forces are present? How can I differentiate between these 2 scenarios: A) the box I am in is rotating or B) I am near an event horizon and the gravitational gradient between masses M1 and M2 is sufficient to stretch the spring
The equivalence principle only applies when the size of the box is small enough that gravitational gradients are undetectable.
^So are we saying it is possible to detect the force of gravity as a free falling observer?
No. Did you read my comment?
Dale, If I understand what you wrote correctly the equivalence principle only applies to a subset of circumstances, but by using a large enough box and sensitive enough equipment, detection of gravitational gradients is in theory possible for free falling observers, for example when gravitational gradients near an event horizon are sufficient to stretch a spring, as was written by Ibix:

We are saying that - with sufficiently precise measurements - it is possible to detect tidal gravity.

#### metastable

If the box is rotating, it's not freely falling. The box's center of mass might be, but the box as a whole is not.
This point makes sense but also raises a separate point of confusion... By the same convention can any classical object be considered "freely falling" when a gravitational gradient is present (ie if different parts or particles of an object experience gravitational gradients, but cohesive forces are holding the particles together, can we say the classical object is freely falling, or does this definition of freely falling only apply to quantum objects)?

#### Dale

Mentor
Dale, If I understand what you wrote correctly the equivalence principle only applies to a subset of circumstances, but by using a large enough box and sensitive enough equipment, detection of gravitational gradients is in theory possible for free falling observers, for example when gravitational gradients near an event horizon are sufficient to stretch a spring, as was written by Ibix:
Yes, but such circumstances are explicitly excluded by the equivalence principle, which you invoked in your opening post. You cannot have it both ways. Either your circumstances can address the equivalence principle, or you can detect gravitational gradients, not both.

Since you are explicitly invoking the equivalence principle in the scenario described by the OP then it is not possible for your springs to detect a gravitational gradient, by definition.

#### PeterDonis

Mentor
By the same convention can any classical object be considered "freely falling" when a gravitational gradient is present
If the object is large enough that tidal gravity is detectable from one end of it to the other, then no, the entire object will not be freely falling, only its center of mass will be.

#### PeterDonis

Mentor
does this definition of freely falling only apply to quantum objects
Including quantum effects doesn't change anything I said. Quantum mechanics just gives a more complete account of how the internal forces inside objects work; it doesn't somehow magically avoid those internal forces when tidal gravity is present.

#### pervect

Staff Emeritus
Dale, If I understand what you wrote correctly the equivalence principle only applies to a subset of circumstances, but by using a large enough box and sensitive enough equipment, detection of gravitational gradients is in theory possible for free falling observers, for example when gravitational gradients near an event horizon are sufficient to stretch a spring, as was written by Ibix:
The way we usually say it is that we say we can detect tidal gravity in the neighborhood of a point. The main question in my mind is if "tidal force" or "tidal gravity" needs further explanation or not. It probably does, but I'm not sure how to do that concisely, though I've tried to talk a little bit about it.

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#### pervect

Staff Emeritus
The way we usually say it is that we say we can detect tidal gravity in the neighborhood of a point. The main question in my mind is if "tidal force" or "tidal gravity" needs further explanation or not. It probably does, but I'm not sure how to do that concisely, though I've tried to talk a little bit about it.
I'm going to reply to myself to expand on my previous point. Consider the moon. It's a massive object and it has gravity, though not as much as the Earth's gravity. Can we detect the Moon's gravity on Earth?

The usual answer is no, not as such, but we CAN detect the tidal gravity of the moon. It raises tides on the Earth. That's basically the point I want to make about the way we usually use the language.

"Measuring Fictitious Forces in a Closed Box"

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