# Equivalence principle question

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mucker
The equivalence principle states that a person stood on Earth would experience “gravity” the same as if he was in an elevator in space traveling at 1g. I get this. but when Einstein was first exploring this, I read he came to the realisation that a person free falling on Earth (if in a vacuum) would have the feeling of weightlessness and therefore is the equivalent of being in an inertial frame. I have read this in a few places but here is a quote form Wikipedia:

From this principle, Einstein deduced that free-fall is inertial motion. Objects in free-fall do not experience being accelerated downward (e.g. toward the Earth or other massive body) but rather weightlessness and no acceleration.

It’s this quote I am struggling with. Firstly, I thought that if you were free falling you are moving at one 1g? therefore you are accelerating, and my understanding is an inertial motion is not accelerating so the statement seems to contradict itself. Secondly, although not based on any scientific grounds, it is not what I have personally experienced when I did a skydive. Ask anyone who has ever done a skydive and they will tell you that when you jump out of a plane, you feel yourself accelerating incredibly fast. Strangely, it’s only when you reach terminal velocity you feel like you are floating (note not weightless but similar), because you are no longer accelerating. This is the point imo where you achieve inertial motion, and that only happens due to drag - which wouldn’t be present in vacuum.

so in summary, imo there are two indicators there when free falling on Earth you are accelerating, which means you can’t be in inertial motion. I am sure it is due to how I have interpreted it, can someone please explain where I am going wrong?

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Say you are in a box of elevator, your feet feel force from the floor ##g\pm a## multiplied by your weight with plus a for accelerating up, minus a for accelerating down, zero a for constant speed including zero. Next say your feet feel zero force from the floor, the box may be free falling to the Earth or while you were sleeping the box was launched to the space and it may be doing inertial motion in space forever. You cannot distinguish them at moment though you will know you continue to travel in space forever or soon die with crash to the Earth. Such local and temporal similarity between inertial motion in IFR and free fall under gravity is the essence of equivalence principle.

Ref: astronauts training making use of this similarity

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Mentor
I thought that if you were free falling you are moving at one 1g? therefore you are accelerating
In discussions like this it is important to distinguish between two different concepts of acceleration. The first is “coordinate acceleration” and the second is “proper acceleration”. Coordinate acceleration is the second derivative of the position, and it depends on the reference frame used. Proper acceleration is what is measured by an accelerometer, and it is frame invariant.

So you are correct that a free falling body is accelerating at 1 g of coordinate acceleration in the reference frame of the ground. However, at the same time an accelerometer would measure 0 g so the proper acceleration is 0.

when you jump out of a plane, you feel yourself accelerating incredibly fast
Your brain does lots of illusions. On the ground you are constantly being proper-accelerated upward at 1 g. This causes pressure on your feet and the auditory canal fluid to pool on the bottom. This is how the body detects acceleration. Like most any sensation, when it is prolonged the brain tunes it out, which means not perceiving those acceleration-stimuli as acceleration. So the sudden absence of those acceleration-stimuli is misinterpreted by the brain as a sudden acceleration instead of as a sudden cessation of previous acceleration. But if you pay attention to the actual stimuli you will see that they are consistent with non-accelerated motion.

The other thing that your brain uses for detecting acceleration is your eyes. But by their nature they can only detect relative acceleration. Essentially a coordinate acceleration. Near the Earth your eyes naturally adopt the frame of the earth, but in space your eyes naturally adopt the frame of the spacecraft

cianfa72, SiennaTheGr8, Imager and 6 others
mucker
So the sudden absence of those acceleration-stimuli is misinterpreted by the brain as a sudden acceleration instead of as a sudden cessation of previous acceleration. But if you pay attention to the actual stimuli you will see that they are consistent with non-accelerated motion.
So what you are saying is that my "perception" of downward acceleration when jumping out of a plane is actually the absence of no longer being proper accelerated upward?

But at the same time I am being accelerated down at 1g in the coordinate acceleration reference? Is another way of looking at it that the Earth is accelerating upward and I am stationary then.

Mentor
So what you are saying is that my "perception" of downward acceleration when jumping out of a plane is actually the absence of no longer being proper accelerated upward?
Yes, particularly the suddenness of it. If you are in free fall for long enough then your brain acclimates and tunes that out too.

But at the same time I am being accelerated down at 1g in the coordinate acceleration reference? Is another way of looking at it that the Earth is accelerating upward and I am stationary then.
Yes, exactly. Coordinate acceleration is relative to each given reference frame.

vanhees71
mucker
Ok thanks, that clears it up! I'm getting there lol! Now I see that in all the articles I read, they were referring to proper acceleration and not the other type - I didn't even know there were two types.

One last question - So whereas speed is ALWAYS relative (as in we don't really know if one object is moving or not), this would be analogous to the cooordinated acceletration? As in we don't really know which object is accelerating (when referencing two objects for example) - but proper acceleration can and actually does tell us if a body truly is accelerating?

Dale
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One last question - So whereas speed is ALWAYS relative (as in we don't really know if one object is moving or not), this would be analogous to the cooordinated acceletration? As in we don't really know which object is accelerating (when referencing two objects for example) - but proper acceleration can and actually does tell us if a body truly is accelerating?
I'd avoid terms like "truly" because they just end up in pointless fights about the nature of truth. But more or less, yes. A better way to put it is that you can detect proper acceleration inside a closed box (you'll hear the term "locally" used for this) by dropping a ball and seeing whether it floats there or starts falling. There is no such test for coordinate acceleration, velocity, or position. For all of those you need to pick some arbitrary object (Earth's surface is a popular choice in every day life) and measure relative to that, which you can't really do from inside a box.

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I take it the “guts in your throat” feeling when you plunge in a roller coaster is what free fall feels like?

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I take it the “guts in your throat” feeling when you plunge in a roller coaster is what free fall feels like?
I don't think roller coasters typically reach free fall, but you are feeling the effects of reduced gravity there. So I imagine the answer is "yes, but more so".

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Ok thanks, that clears it up! I'm getting there lol! Now I see that in all the articles I read, they were referring to proper acceleration and not the other type - I didn't even know there were two types.
This is one reason it's advantageous to know some classical physics before studying relativity. Newton's second law involves force and acceleration: $$\vec F = m\vec a$$ Why acceleration, why not position or velocity? The reason is that position and velocity are frame dependent. There's no point that is always the origin; and, there is no way to ascribe a specific, universal speed to an object (that idea goes back to Galileo). In other words, an object has no universal position or velocity

So, absolute acceleration is the first thing that is measurable. In principle and in practice, you can measure the force on something and calculate its absolute acceleration.

Suppose, however, you are in a car that accelerates away from your house. From the reference frame of the car, the house clearly accelerated away. This was not the result of a force on the house, but because you measured the position of the house from an accelerating reference frame.

In Newtonian physics, you have to distinguish these two cases:

Acceleration in an inertial frame is the result of (real) forces on the object.

Acceleration in a non-inertial frame may be the result fictictious forces - and this concept generalises to coordinate acceleration.

Examples of fictitious forces include the centrifugal force associated with circular motion and the Coriolis force:

https://en.wikipedia.org/wiki/Coriolis_force

vanhees71 and dextercioby
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Ok thanks, that clears it up! I'm getting there lol! Now I see that in all the articles I read, they were referring to proper acceleration and not the other type - I didn't even know there were two types.

One last question - So whereas speed is ALWAYS relative (as in we don't really know if one object is moving or not), this would be analogous to the cooordinated acceletration? As in we don't really know which object is accelerating (when referencing two objects for example) - but proper acceleration can and actually does tell us if a body truly is accelerating?
In short, yes. I also would use a neutral and scientific term like “invariant” rather than “true” which is more emotionally loaded.

Physicists like invariant quantities. The idea is that coordinates are not part of nature, they are an arbitrary part of our analysis. So the feeling is that you get more insight about nature by focusing on things that do not depend on that arbitrary coordinate system.

But on the other hand, coordinate quantities are very useful, and include things like energy which is almost unavoidable. Also, properly choosing your coordinates can mean the difference between an unsolvable and a solvable problem. So frame-variant quantities are important too, which is why it is best to avoid loaded language.

It is important to distinguish between invariant and variant quantities, but we will use them all.

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mucker
I'd like to expand on another idea I've had after reading and taking all this in.

If when we are in free fall, we are in an inertial frame and therefore not accelerating then the logical conclusion is that the Earth accelerating towards us (I also read this somewhere). But how can the Earth be constantly accelerating towards two skydivers on opposite sides of the planet? It's like the Earth would be coming apart. Now I know this is all in relation to that the mass of the Earth curves spacetime but can someone (I know it will be difficult) please explain in as simple as possible how this works?

vanhees71 and PeroK
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I'd like to expand on another idea I've had after reading and taking all this in.

If when we are in free fall, we are in an inertial frame and therefore not accelerating then the logical conclusion is that the Earth accelerating towards us (I also read this somewhere). But how can the Earth be constantly accelerating towards two skydivers on opposite sides of the planet? It's like the Earth would be coming apart. Now I know this is all in relation to that the mass of the Earth curves spacetime but can someone (I know it will be difficult) please explain in as simple as possible how this works?
When we have curved spacetime inertial frames are only valid locally. There are no global inertial frames in GR (General Relativity). The relative motion of an object falling to Earth and the local surface of the Earth can be analysed (approximately) using the local inertial reference frame associated with freefall.

You cannot, however, analyse the entire planet Earth from that frame: the local inertial coordinates are simply not a good approximation beyond a small portion of spacetime.

vanhees71
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Re: post #12

One skydiver has his own frame of reference and another skydiver has her own frame of reference. They do not coincide but explain the events in good accordance.

For an example,
He observes the Earth coming to him with some acceleration and in its backside her coming with more acceleration.
She observes the Earth coming to her with some acceleration and in its backside him coming with more acceleration.

mucker
I don't get either of those explanations, at least not in regards to the question I asked (I don't have any physics background at degree level but I am learning more each day!). So if you were to chose one, would you say the person is accelerating down, or the Eather accelerating up? I do not believe this is relative to each other due the description of what "proper" acceleration is above which Dale explained.

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Please find attached the sketch for my post #12. I hope it may help us for good communication.

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I don't get either of those explanations, at least not in regards to the question I asked (I don't have any physics background at degree level but I am learning more each day!). So if you were to chose one, would you say the person is accelerating down, or the Eather accelerating up? I do not believe this is relative to each other due the description of what "proper" acceleration is above which Dale explained.
An object on the surface of the Earth is subject to a force and hence proper acceleration. The object in freefall is subject to no force and has no proper acceleration. When measured from the accelerating reference frame of the Earth's surface, the falling object is accelerating downwards. Compare the house and car example above.

This is an I level thread, which assumes some knowldege of concepts such as local and global reference frames.

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So if you were to chose one, would you say the person is accelerating down, or the Eather accelerating up?
Either is fine locally (i.e. as long as you don't think about skydivers on opposite sides of the Earth). But globally you end up with something like "you are (coordinate) accelerating towards the Earth".

The fun bit is that the Earth's surface is genuinely undergoing proper acceleration outwards. It doesn't move anywhere, though, because we're in curved spacetime, and acceleration doesn't necessarily lead to movement except in flat spacetime...

mucker
anuttarasammyak I understood your point first time but thanks for taking the time to hand draw a diagram :-)

PeroK I do understand you and what Annuta said, I think it's that you didn't understand my question. You were giving explanations about something I never asked :-) My point was, if for example I skydive at north pole, then the Earth is proper accelerating in that direction, but if another person was skydiving from south pole then the Earth is proper accelerating to them too. The Earth can't be accelerating in opposite directions at the same time otherwise it would explode...Or… can it? That is what I was asking to be explained. And that is what Ibix just did. I knew it was related to spacetime curvature but couldn't get my head around, still can't to tbh but I get it now... kinda.

The fun bit is that the Earth's surface is genuinely undergoing proper acceleration outwards. It doesn't move anywhere, though, because we're in curved spacetime, and acceleration doesn't necessarily lead to movement except in flat spacetime...

It is almost as if the Earth is accelerating outwards constantly but then coming back in on itself (only way to explain why it doesn’t explode), like it’s going inside out (maybe it is, and this due to the spacetime curvature which keeps it together) . Interestingly I just thought that this is exactly how a 4 spatial dimensional object behaves when it rotates about it's 4th axis (see hypercube animation here).

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If you want to understand GR you need to find a way to stop thinking entirely in terms of classical physics. Even SR is explicitly incompatible with classical notions of space and time.

Note that I fully understood your question.

Each point on the Earth's surface is accelerating outwards. But, acceleration is not motion. And, in curved spacetime, that continuous acceleration does not lead to any change in the size or shape of the Earth.

To analyse the entire globe you need a global frame of reference and that entails including curved spacetime. You can't pretend the entire spacetime is flat, anymore than you can pretend the surface of the Earth is flat. Even though, locally, it may be flat to a good approximation.

This is the price for abandoning gravity as a force. You can't just give up the gravitational force and carry on with Newtonian concepts. If you give up the gravitational force, you must adopt general relativistic spacetime.

Locally, by choosing inertial coordinates you may continue to apply a somewhat classical description of a falling object. But, that only works locally. Each skydiver may use their own frame locally, but these local frames do not extend to encompass each other.

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Do you even need GR to find an example where two observers feel a force in opposite directions but remain at rest wrt each other? If two observers are in a rotating spaceship, but on opposite sides, should they not both measure a proper acceleration, but which is in opposite direction to each other’s?

And if so, and it’s unsurprising there, it shouldn’t be too unsurprising to see the GR version of it in this Earth problem.

mucker
PeroK thanks! And I am working on understanding it! Hence all these questions :-). On a side note, I've decided to do a degree in Physics so can understand it fully with the math as well; this has been something I've interested in for decades but only really started to put serious effort into lately.

On another note, I just watched a good YT video which visualises GR better than any other I've seen. In the past few weeks with all the reading I've done on GR and spacetime curvature, what I was struggling to understand (and I guessed I assumed it) was that spacetime is curved... but I think using the word "curved" is a bit misleading. II would say a better of visualising it is that it is 'sucking' in spacetime (I know that's a poor description but you'll see my point next), it makes it easier to visualise and comprehend. I have been thinking this for weeks now as i read more and more, this is how I visualise it after all the reading (like how i surmise it in my head to grasp it). Then interestingly I find this YT video that shows exactly how I visualise it. Would you say this is a good way to visualise it? It then makes sense how the Earth can accelerate in all directions without moving (they cancel each other out). If the YT video is a good way to visualise spacetime curvature (is it?) then this is what I mean by the nomenclature used to describe spacetime being "curved" is misleading; a more apt description would be to use the word "sucking/pulled/dragged" in there somehow (I can't think of a good word describe it but all these fit it better than "curved").

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"Sucking, pulling, dragging" is misleading. Curved in the sense of differentiable manifolds is correct.

The spacetime around the Earth is essentially static.

mucker
"Sucking, pulling, dragging" is misleading. Curved in the sense of differentiable manifolds is correct.

The spacetime around the Earth is essentially static.
But isn't it then that if the Earth is constantly accelerating outward (and therefore in flat spacetime would move), the fact that it doesn't move (as Ibex said above), mean that the spacetime is constantly being pulled/dragged into the centre of the Earth to counteract the outward acceleration? Or at least in the case of Earth where the gravity is strong?

Most of what I am saying will make sense where I am coming from (even if wrong) if you look at that YT vid. If you get chance I'd really appreciate it if you wouldn't mind checking it out and telling me if it's a good way to visualise it, so I know if it's wrong or not.

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You're inventing dynamic physical phenomena in order to come to terms with the nature of static spacetime. Look at @Grasshopper 's post, for example. Uniform circular motion has continuous acceleration always orthogonal to the direction of motion.

Moreover, in the rest frame of an object in uniform circular motion, there is a measurable unbalanced force but no motion. You don't need to invent an oscillation to explain this. The concept of an accelerating reference frame explains it simply and adequately.

The same is true for the surface of the Earth in the GR model of curved spacetime.

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That video got reviewed on here recently. You could try to find the thread.

mucker
haha I foudn the thread. Seems it's quite wrong from what people are saying. This is the problem with the Internet!

Mentor
But how can the Earth be constantly accelerating towards two skydivers on opposite sides of the planet? It's like the Earth would be coming apart.
I wrote an Insights article on this topic: https://www.physicsforums.com/insights/understanding-general-relativity-view-gravity-earth/

Basically, what you say would be correct in flat spacetime, but not in curved spacetime. First, we need to understand the idea of spacetime. Instead of having 3D space that changes over time we consider a 4D spacetime where time is just another direction (time is the 4th dimension). A single particle traces out a line in spacetime, called a worldline.

Now, in this view, an inertial particle has a straight worldline, and an accelerometer measures how tightly a worldline turns and what direction it turns. Also, in this view, two particles that are at rest with respect to each other have parallel worldlines.

So, let's consider the skydivers and remove the atmosphere. Their accelerometers read 0 so their worldlines are straight (geodesic). They are initially at rest with respect to each other, so they are initially parallel. Now, in standard flat Euclidean geometry, two lines that are both straight and initially parallel will be parallel everywhere and the distance between them will not change. But consider a sphere which is a curved surface, on a sphere the straight (geodesic) lines are great circles, e.g. longitude lines. So consider two nearby longitude lines starting at the equator and going north. They are straight, and initially they are parallel (at the equator), but as you go further north they are no longer parallel and eventually intersect (at the pole).

So curvature allows two parallel straight lines to change the distance between them and even intersect, which is not possible in a flat space. Similarly, curvature also allows two curved lines to continually curve away from each other without the distance between them changing. This is why the surface of the Earth can accelerate upward without expanding.

I recommend reading the Insights for a fuller treatment.

DAH, PeterDonis and vanhees71
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haha I foudn the thread. Seems it's quite wrong from what people are saying. This is the problem with the Internet!
Even if those videos are good, IMO the problem is that they try to create understanding in the wrong way. They provide some animation that changes the new theory to conform to your preconceptions and classical modes of thinking.

Instead, real understanding can only come from changing your own way of thinking. For GR, that starts with learning SR and a radically new theory of spacetime. That's hard and takes perhaps a couple of months of study. And, ultimately, you need to develop the mathematical skills to do actual problems.

GR is an advanced undergraduate subject. There is no easy way to get a firm grasp of it, let alone master the subject. No video is going to do that.

martinbn and PeterDonis
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My point was, if for example I skydive at north pole, then the Earth is proper accelerating in that direction, but if another person was skydiving from south pole then the Earth is proper accelerating to them too. The Earth can't be accelerating in opposite directions at the same time otherwise it would explode...Or… can it?

All the skydivers and the Earth have zero proper acceleration.

Ref. Wikipedia proper acceleration https://en.wikipedia.org/wiki/Proper_acceleration

In relativity theory, proper acceleration[1] is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured. Gravitation therefore does not cause proper acceleration ...

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All the skydivers and the Earth have zero proper acceleration.
You mean the Earth in its orbit around the Sun?

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I have not included the Sun. But even including the Sun as gravitation does not cause proper acceleration I suppose the Earth keeps zero proper acceleration.

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But even including the Sun as gravitation does not cause proper acceleration I suppose the Earth keeps zero proper acceleration.
The Earth as a whole has zero proper acceleration on average, yes (...I'd better cover myself and say I'm handwaving a bit here because tidal gravity is a thing and I haven't completely formalised the averaging process). But the surface, and any region inside except at the very center, does have proper acceleration outwards from the center, just to be clear

anuttarasammyak, Dale and vanhees71
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All the skydivers and the Earth have zero proper acceleration.
You have to be careful here. The surface of the Earth has non-zero proper acceleration upwards, as does almost every part of the interior of the earth. Only the very center of the Earth has zero proper acceleration.

Edit: I see that @Ibix already made the same point, and better!

vanhees71
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Re: post #12

One skydiver has his own frame of reference and another skydiver has her own frame of reference. They do not coincide but explain the events in good accordance.

For an example,
He observes the Earth coming to him with some acceleration and in its backside her coming with more acceleration.
She observes the Earth coming to her with some acceleration and in its backside him coming with more acceleration.
This is wrong, because neither skydiver's frame covers a large enough region of spacetime to include the other skydiver (or even the whole Earth--only a small portion of the Earth's surface nearest to the skydiver is covered). So neither skydiver can attribute "more acceleration" to the other skydiver than to the nearest portion of the Earth's surface.