# Inertial frame of freely falling body

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• bksree

#### bksree

A freely falling body (falling in earth) accelerates with time. However, an object inside the body remains at rest (if it was initially at rest) or moves with a constant velocity if it was initially moving with constant velocity. In other words a frame fixed to the body is an inertial frame. But since the frame is fixed to an acceleating bod, it is also accelerating. Then how can it be an inertial frame ?

Can someone clear this confusion.

Dale

Can someone clear this confusion.
The freely body is only accelerating if you choose to consider the surface of the Earth to be at rest, and the frame in which the surface of the Earth is at rest is not an inertial frame.

This situation is confusing when described in plain language because the word "acceleration" is ambiguous: It can refer either to proper acceleration, which is what an accelerometer measures and can be measured without reference to any external bodies; or to coordinate acceleration which describes the change in a body's speed relative to some arbitrarily chosen external reference point like the surface of the earth.

The proper acceleration of the falling body, an object inside of it, and an object freefalling alongside it are all zero, and all of these objects can be considered to be at rest in an inertial frame in which the surface of the Earth is moving towards them. The proper acceleration of the surface of the Earth is 1G upwards (an accelerometer placed on the surface of the Earth will confirm this fact) so the frame in which the surface of the Earth is at rest is not inertial.

In the non-inertial frame in which the surface of the Earth is at rest, the falling object has non-zero coordinate acceleration; in the inertial frame in which the falling object is at rest the surface of the Earth has non-zero coordinate acceleration.

The key to understanding general relativity is to break yourself of the habit of attaching any importance to coordinate acceleration - you can make it come out to be whatever you please just by choosing a different coordinate system.

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cianfa72, nitsuj, PeterDonis and 2 others
Can someone clear this confusion.

I believe your confusion is totally justified. According to Newtonian physics the freely falling body cannot be an inertial system(assuming the Earth is). It is not inertial because it accelerates with respec to to absolute space.
Your confusion has led Einstein to postulate the equivalence principle(EP). He thought the theory of gravitation based on EP would solve the mistery of abslute space, however it seems that it didn't.

He thought the theory of gravitation based on EP would solve the mistery of abslute space, however it seems that it didn't.

Why do you say it seems that it didn't?

I believe your confusion is totally justified. According to Newtonian physics the freely falling body cannot be an inertial system(assuming the Earth is).

His confusion is not justified, because he is asking this question about relativity, not Newtonian mechanics. In Newtonian mechanics it is true that a body freely falling in the Earth's gravitational field is not at rest in an inertial frame. But the definition of "inertial frame" is different in GR than in Newtonian mechanics. And, as @Nugatory has pointed out, the GR definition has the huge advantage of corresponding to a direct physical observable (what is measured by an accelerometer). With the GR definition, there is no confusion.

vanhees71
Why do you say it seems that it didn't?
Because he thouht he could get rid of abslute space in the sense of what he called Match principle. I'm not an expert but according to them GR failed to include Macht principle

His confusion is not justified, because he is asking this question about relativity, not Newtonian mechanics. In Newtonian mechanics it is true that a body freely falling in the Earth's gravitational field is not at rest in an inertial frame. But the definition of "inertial frame" is different in GR than in Newtonian mechanics. And, as @Nugatory has pointed out, the GR definition has the huge advantage of corresponding to a direct physical observable (what is measured by an accelerometer). With the GR definition, there is no confusion.
GR with its equivalence principle only states that gravity is equivalent to acceleration. Of course this means that a freely falling object is locally an inertial frame.
However this does not release us from the background "absolute space". Again, according to historian and experts, at the beginning Einstein thought his theory could include as a consequence Match principle thus explaining absolute space or rather releasing us from it.

A freely falling body (falling in earth) accelerates with time. However, an object inside the body remains at rest (if it was initially at rest) or moves with a constant velocity if it was initially moving with constant velocity. In other words a frame fixed to the body is an inertial frame. But since the frame is fixed to an acceleating bod, it is also accelerating. Then how can it be an inertial frame ?

Can someone clear this confusion.

It can be explained in this way: According to the equivalence principle a frame fixed on Earth surface cannot be inertial because it is equivalent to and accelarated system with accelaration a=-g where g is equal to Earth gravitatinal field. So there is no problem with the freely falling body being an inertial frame because it accelates with respect a non inertial system.

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he thouht he could get rid of abslute space in the sense of what he called Match principle. I'm not an expert but according to them GR failed to include Macht principle

The question of GR and Mach's principle is much more complicated than you appear to realize. There is at least one viewpoint according to which it is obvious that GR satisfies Mach's principle: the source of gravity in the Einstein Field Equation is the stress-energy tensor, nothing else. That means that the geometry of spacetime--what we think of as "gravity" and "inertia"--is determined by the distribution of matter and energy, just as Mach's principle says. Cuifolini & Wheeler wrote an entire textbook, Gravitation and Inertia, on this topic.

GR with its equivalence principle only states that gravity is equivalent to acceleration.

This is a very vague and possibly misleading way of stating what the EP says.

Of course this means that a freely falling object is locally an inertial frame.

A freely falling object is at rest in a local inertial frame.

this does not release us from the background "absolute space".

Yes, it does. See my comments above on GR and Mach's principle. You really need to get more background in this area before making pronouncements.

A freely falling body (falling in earth) accelerates with time. However, an object inside the body remains at rest (if it was initially at rest) or moves with a constant velocity if it was initially moving with constant velocity. In other words a frame fixed to the body is an inertial frame. But since the frame is fixed to an acceleating bod, it is also accelerating. Then how can it be an inertial frame ?

Can someone clear this confusion.

Strictly speaking, in the presence of gravity, neither the frame falling with a body nor the frame at rest is an inertial frame. In the free-falling frame, for instance, there are tidal forces.

The general relatistic view is essentially that the free-falling frame is the simplest frame to describe, and that the tidal forces in this frame wind up representing gravity. With some rather beatiful but advanced math, these tidal forces can be equated to the abstract concept of curvature. The presence of space-time curvature means that neither frame near a large mass is an "inertial frame", though.

The Newtonian view considers the frame attached to the Earth to be inertial (and ignores some experimentally observed effects like gravitational time dilation to do so). The Newtonian view is essentially an approximation from the point of view of GR, though under the right circumstances it can work reasonably well. The Newtonian view will only work as an approximation, if one becomes interested in gravitational time dilation for instance, the Newtonian view will not suffice, the Newtonian view does not treat this coherently. . In the Newtonian view, gravity is a real force.

In the GR view, gravity is the tidal force, and what we call "gravity" in the Newtonian point of view is more akin to a fictitious force due to the acceleration. Though the fictitious force model isn't quite rich enough, really.

This is drastically oversimplifed, but it's the best I can do without some rather advanced math.

You really need to get more background in this area before making pronouncements.
Sorry I did not know that this forum was open only to experts. I thought it was a place to respectfully interchange opinions, ask questions and learn.

The forum is certainly open to other than experts, it's not intended to just have experts talking to themselves. However, there is more structure here than everyone (regardless of knowledge) expressing their opinion and expecting the reader to sort out the right opinion from a vast array of competing opinions (the free speech model).

It might be helpful to look at the Physics Forums global guidelines <<link>>.

The mission statement is:

Our mission is to provide a place for people (whether students, professional scientists, or others interested in science) to learn and discuss science as it is currently generally understood and practiced by the professional scientific community. As our name suggests, our main focus is on physics, but we also have forums for most other academic areas including engineering, chemistry, biology, social sciences, etc.

To this end, some effort to know how physics is "generally understood and practiced" is expected by particpants answering questions, the standard is higher than "this is my opinion and I'm sticking to it". If there is a disagreement on this issue, discussion is certainly welcome. References to the literature and textbooks are highly encouraged in such disagreements to help resolve the disagreements.. If someone is seeking an answer to a question, and a third party comes in expressing an "opinion" that is seen as conflicting with how physics is generally understood, it becomes an obstacle to the mission statement. Probably the best thing to do is to hash the issues out, politely and professionally, in another thread, to avoid hijacking the original thread.

.

dextercioby
I thought it was a place to respectfully interchange opinions, ask questions and learn
Even so I think a place where people throw out their opinions without checking whether or not they are in agremment with current theories, makes it difficult if the goal is to get a coherent discussion and answer participants questions. Remember that science should be self consistent and that means being consistent with your arguments about scientific theories.

Sorry I did not know that this forum was open only to experts. I thought it was a place to respectfully interchange opinions, ask questions and learn.

If you ask questions, that's one thing. If you make statements, and they are not accurate, or are potentially misleading, that's something different. You should expect to be corrected if you do the latter, not because only experts can make statements, but because inaccurate or potentially misleading statements by anyone here, expert or not, need to be corrected for the benefit of all of our users.

@kent davidge made a good observation on this:

I think a place where people throw out their opinions without checking whether or not they are in agremment with current theories, makes it difficult if the goal is to get a coherent discussion and answer participants questions.

Nugatory, Peter Donis, Facenian, Pervect, Kent Savidge
Thank you for your responses. I am an engineer with no previouse knowledge of GR. Recently I stumbled on the book 'Relativity Visualised' by Epstein and took up the journey of self studying GR.
I have started wih 'Spacetime Physics' by Taylor and Wheeler. In the very first chapter I came across the float frame idea and stopped in my tracks.
I've got a vague idea now and I am starting again.

Thank you

Taylor's & Wheeler's "Space-time physics" is an excellent book, giving the modern treatment of special relativity. I'm not familiar with Epstein's book, but I gather it's pretty popular. For a basic introduction, I'm rather fond of Bondi's "Relativity and common sense", but if you have the background for "Space-time physics" it's a much-more indepth and advanced book with many insights.

A freely falling body (falling in earth) accelerates with time. However, an object inside the body remains at rest (if it was initially at rest) or moves with a constant velocity if it was initially moving with constant velocity. In other words a frame fixed to the body is an inertial frame. But since the frame is fixed to an acceleating bod, it is also accelerating. Then how can it be an inertial frame ?

It is not an inertial frame in the full sense of the term in special relativity, which extends over all space and time, and in which mathematical terms called Christoffel symbols are zero, and the derivatives of the Christoffel symbols are also zero everywhere.

In the frame attached to a freely fallling body, the Christoffel symbols are zero at the origin attached to the body, which is why this is called a local inertial frame. However, the Christoffel sumbols are not zero away from the origin, and the derivatives of the Christoffel symbols are not zero even at the origin, which is why it is not an inertial frame in the full sense of the term in special relativity.

See also pervect's post #10, which makes the same point. He refers to "tidal forces" which are essentially the same thing as the derivatives of the Christoffel symbols.

https://arxiv.org/abs/0806.0464
An electromagnetic perpetuum mobile?
Øyvind Grøn, Sigurd Kirkevold Næss
"The principle of equivalence has a local character. The mentioned equivalence is only valid as far as the measurements does not reveal a possible curvature of space"

https://arxiv.org/abs/1102.0529
The motion of point particles in curved spacetime
Eric Poisson, Adam Pound, Ian Vega
See Equations 9.14 to 9.16 and the brief comment following.

vanhees71