## Pseudoforces and Geometry (or) BigBang and BlackHoles

### February 24, 2006

In mechanics, the concept of a pseudo-force is a useful one. It allows you to work in frames which are non-inertial and apply there the usual laws of Newton. But, one thing about pseudoforces which is rarely discussed, is the fact that pseudoforces tell you something about the geometry of the space around you.

One has to wait till one learns general relativity to recognise this fact. And even then, it is quite likely that this fact is missed in the rain of tensor indices and four vector algebra :)

But the relation between pseudoforces and geometry is a simple one, independent of relativity or gravity. And this relation between pseudoforces and geometry is better discussed first in the familiar Newtonian framework.

Sidenote : Once you understand how pseudoforces affect geometry, the relation between geometry and gravity- the core idea of GR is self-evident. According to Einstein’s equivalence principle, gravitational forces are indistinguishable from pseudoforces and if pseudoforces affect geometry, gravitational forces should do so too !

Well, the beauty of the equivalence principle is that by identifying gravitational forces with pseudoforces it connects a lot of seemingly unrelated facts into a beautiful theory. All this factsabout pseudoforces, geometry and gravity fall into place in General relativity which combines all of them by identifying gravitational forces with pseudoforces and making matter affect geometry via gravitational/pseudo-forces

Now, coming back – how do we see that pseudoforces affect geometry, and yeah why haven’t we seen it before when we learnt mechanics ?

I’ll answer the second question first. We haven’t seen it before, because the usual examples chosen for non-inertial frames – accelerating/rotating frames have such a familiar geometry that we tend to overlook the effect of pseudoforces. The geometry of space is same as the usual High-school geometry which applies to inertial frames , so there is not much new to talk about here. Once we realise this, the question how do we see the effect of pseudoforces on geometry has a straightforward answer : Think about more general non-inertial frames !

Sidenote : By the way, if you take Einstein’s relativity into account, you would see that this is not true – you can prove that there is no non-inertial frame which has the same geometry as the highschool geometry. In particular, in accelerating and rotating frames we would see two additional geometric effects- gravitational redshift and Sagnac effect due to pseudoforces. That indeed is the reason why in relativity, you are forced to acknowledge the effect of pseudoforces on geometry, whether you like it or not.

Now, what do I mean by non-inertial frames more general than accelerating/rotating frames ? Consider, for example, what I call an “expanding” frame – a frame of observers who when seen from an inertial frame are expanding radially outwards from the origin.The frame is defined by the usual convention that according to these set of observers, they are all at rest with respect to each other. If you think about it a bit, you will see why this frame is different from the usual non-inertial frames .

In the case of accelerating/rotating frames, the corresponding observers are in a rigid-body like motion. In particular, this means that the distances between the observers does not change. But, for an expanding frame, the distance between the observers increases with time. It is very simple to convince oneself of this fact- just think of two diametrically opposite observers. The distance between them increases with time. And since distances are absolute in Newtonian framework, these observers will actually see the distance between them increasing, though they are at “rest” with respect to each other (by definition) !

I like this particular example very much, since it brings into a sharp focus our conception of what do we mean by being at rest . If you say that being at rest with respect to some frame means not moving, then these two observers are not moving in the “expanding” frame – they are at rest due to the very way the “expanding” frame has been defined. And the distance between the two observers can still increase, though neither of them is “moving” in this sense ! So, how do the expanding frame’s observers interpret this weird thing they observe ? There is only one way out – in that frame, the space itself is expanding ! Though they are at rest, the space is continuously “created” in between them resulting in the increase of distance.

But, what causes this “creation of space” in this non-inertial frame ? What is that entity which is responsible for the creation of space ? Well, that is easily answered. Remember this golden rule about pseudoforces- “Whatever weird is happening in non-inertial frame, can eventually be blamed on the pseudo forces” ! So, it is the pseudoforce fields that create space.

You can convince yourself by calculating the pseudoforces due to this expansion and by showing that the expanding observers can predict the rate of creation of space by measuring the pseudoforces in this frame. So here we have the first example of effect of pseudoforces on geometry- there are some pseudoforce fields which can create space and increase the distance between the two objects which are at rest. In fact, they can do the reverse thing too, they can “eat up” or “swallow” the space in between two objects at rest.

So in this sense, non-inertial frames are like the wonderlands of Lewis Caroll. If you’ve read “Alice in the Wonderland” you would remember this scene in which the red queen and Alice start running

The most curious part of the thing was, that the trees and the other things round them never changed their places at all : however fast they went, they never seemed to pass anything. …
Alice looked round her in great surprise. Why, I do believe we’ve been under this tree the whole time! Everything’s just as it was!’

Of course it is,’ said the Queen, what would you have it?’
Well, in our country,’ said Alice, still panting a little, you’d generally get to somewhere else — if you ran very fast for a long time, as we’ve been doing.’

A slow sort of country!’ said the Queen. `Now, here, you see, it takes all the running you can do, to keep in the same place.

If you want to get somewhere else, you must run at least twice as fast as that!’

Expanding-frame is a bit like that. You’ve to run in this frame to be at the same distance from something at rest in this frame !

And once you realise this, you are on your way to understand the philosophy behind GR- that space is not a quite thing sitting silently allowing all other things to move. It is not a dead-silent stage for the other actors. The stage called space itself is an actor in the drama. It can be created and eaten up, squished and stretched by the pseudoforces.

In fact, there are only two more steps from here to GR. First, is the realisation that inertial frames need not exist- a spacetime where there is no frame in which pseudoforces vanish everywhere is said to be a “curved” spacetime. So, this realisation can be rephrased as the fact that the spacetime can be “curved”. This terminology comes from the fact that for example, a two-dimensional being over a sphere cannot establish a frame over the sphere in which all the pseudoforces vanish. So, this is one way of defining what do you mean by a curved space-time – you are in a curved spacetime if you cannot establish a single inertial frame everywhere.

The universe we live in is such a space. There are pseudoforce fields that extend all over the universe creating new space – this is what we mean by the expansion of the universe. But unlike the example before, this expansion cannot be gotten rid of. The pseudoforce fields cannot be transformed away everywhere by a change of frame since there are no global inertial frames. And in fact, you can use this to tell about the past of our universe- in the past when all this space was not yet created by the pseudoforce fields, the universe was small and hot, and this is what the BigBang theory is all about.

And this should clear the confusing question “where did BigBang happen ?” – you see when bigbang happened there was only one place where it could happen – the rest of the space was simply not there !

And then there are people who think everything in the universe expands – you know from our measuring rods to earth-sun distance to distance between the galaxies. Then they get confused about how we can see the expansion if everything expands. Well, I just tell them – think of this expanding frame and imagine what those non-inertial guys would see. Equivalence principle assures you that what we see in an expanding universe would be very similar.(The only difference is that the pseudoforces cannot be wished away bya frame choice- the efffect of pseudoforces on geometry are very similar.). It is immediately obvious that the measuring rods these people carry would not expand. True there will be a tendency to expand- there will be stresses developed on the rods due to the pseudoforces for example ! But, they won’t expand appreciably unless the pseudoforces are really strong .

Second, is of course the equivalence principle- There is no way to distinguish gravitational forces from the pseudoforces in a general spacetime. So, they are just two different names of the samething. This in turn, means that matter can produce pseudoforce fields which can then eat up space. How cool !

This gives rise to really cool objects called Black Holes ! What happens here is that the matter produces a pseudoforce field which eats up space at a large rate. In fact, it eats up space so fast that invariably any two things inside a blackhole are brought near each other. The rate of eating of space is so high that you can’t even run away ! Remember – there is a limit to how fast you can run since you cannot run faster than light. There is no escape from the interior of a blackhole because eventually the pseudoforces will eat up the space between any two objects and everything will be crushed together. Like the expanding universe case, these pseudoforces which create blackholes cannot be transformed away or gotten rid of by changing frames.

So, in GR you have this picture of space which is active and alive , with birth and death ! The space is like a deep sea on which all ships stand. When the sea is calm and serene, the explorers aboard can delude themselves that they are on the land – That there is a ‘ground’ beneath their feet, that their life is lived over a rock-steady stage. Oh explorers! how far would you delude yourselves ? When the sea comes to her full glory, When her waters drench your clothes, When the platform on which you stand begins to shake, would you still mainatain that you are on land ? Would you still dismiss its dynamical nature ? That, in a way, is the challenge of gravity to anyone who seeks to understand her.

P.S. : Do have a look at this article titled Albert Einstein’s Theory of Relativity- In Words of Four Letters or Less . It talks about the same topic but in a different(and probably better) way.

### 11 Responses to “Pseudoforces and Geometry (or) BigBang and BlackHoles”

1. bharath Says:

give me some example where I can see a pseudo force. Suppose a body is moving faling from a mountain. I observe it from a frame of reference that itself is falling at a lower rate of acceleration. then what should I see? what is difference if it is not accelerating.

the article seems little too abstract. it would be nice to know the objects about which you are talking. at least for the non experts.

2. nayagam Says:

Hi Bharath,

give me some example where I can see a pseudo force.

Well, Pseudoforces are there everywhere ! Say, you are sitting in a bus/car and say the driver applies a brake causing the moving vehicle to stop suddenly – Imagine what would it feel like ? It would feel like something is pushing you – you would feel a force. But, if you just look around, there is NOTHING really there pushing you ! It is as if there is some invisible ghost which is producing this force – hence, the name “pseudo-force”(literally false-force or fake-force). If you will, it is the same “ghostly” force which you feel when you are sitting on a vehicle which takes a sharp turn.

It is an amazing fact about our world and it is a real weird thing, if you come to think of it ! (In fact, historically, Mach was among the first to remark on how weird this whole thing is. And,Einstein was motivated a lot by Mach’s ideas when he was putting together general relativity.)

3. bharath Says:

thats the D’Alembert principle! does inertial and non-inertial frames mean stationary and moving frames of reference? like a frame stuck to a billboard as opposed to one stuck to a moving car?

then what is the change in geometry due to this pseudo force? I can see that the world appear to behave (move) differently when we go from a stationary to a moving frame of reference. is that it?

when you mean an expanding frame, you attach a distance measure to the frame which changes as the frame expands, so that two points are always at a contant distance with respect to the frame. but for an observer on the bill-board (say) the points will be receeding from each other? if this understanding is right, then why is it natural (any intuitive reasons?) to choose a distance measure depending on the frame of reference.

I think I am slowly getting to the hard part of the blog now. thanks for keeping up the patience.

4. nayagam Says:

Suppose a body is moving faling from a mountain. I observe it from a frame of reference that itself is falling at a lower rate of acceleration. then what should I see? what is difference if it is not accelerating.

Ha ! Great! This question in fact goes into the heart of Equivalence principle and the crucial difference between how Newton and Einstein look at the same thing! I didn’t want to go into this in this post, since equivalence principle demands a separate post altogether. But, now that you’ve asked, let me try to explain as briefly as possible.

If you don’t mind, can I make it a ball dropped from a tall tower ? Say, let us start with the case when you are standing at rest relative to earth’s surface. Ok, now since the days of Galileo, we have known what happens to a ball dropped from a tower on earth. As it comes down, it moves more and more fast- it accelerates. And that too at a steady-rate of g=9.8 metres/second-squared, i.e, for every single second the ball spends falling, the speed of the ball increases by 9.8 metres/second.This tells us immediately via Newton’s laws that there is a force acting on the ball. (In fact, we know the name of this force-“Weight” :) )Now, the million dollar question is this -Is this a pseudoforce ?

Now, that’s a tricky question. In fact, it’s so tricky that Newton answered it wrongly ! Newton thought that this is not a pseudoforce, because there is earth’s gravitational field to take the blame for applying force on the ball. So, this is supposed to be “real” force and not a “fake”-force.

Now, enter Einstein ! Einstein brought with him what is called the equivalence principle – there is no way to distinguish between the “real” forces produced due to gravitation and the “fake” pseudo-forces, so they are one and the same. Another way of putting it is as follows- You always measure only the sum of the “real”+”fake” forces, and there is no experiment you can design which can separate them(according to Equivalence principle) – so this very division of forces into gravitational forces vs pseudoforces is meaningless according to Einstein ! So, under this kind of philosophy, what is acting on the ball can justifiably called pseudoforce.

And the story is same for each kind of frame you may care to imagine. Newton goes to every frame, looks at the force and almost arbitrarily divides them into “real¨ vs “fake” whereas Einstein looks at the whole force and calls it the one and the only one gravitational pseudoforce.

5. bharath Says:

loved the alice in wonderland example. so the expading frame is like rubber like frame thats getting stetched due to pseudo forces (‘invisible forces’)

okie. so what is spacetime? As I understood it, the frame of referece only changes the way space gets represented. time appears to be a common denominator with no contraint on it.

6. bharath Says:

I will let you in peace and let you blog this in your leisure. so you can do justice to it, and it might also benefit others. :) what say?

I pestered you because your blog starts with topics I understood decently, but then I couldn’t quite see how it all worked out as you describe in your blog.

7. nayagam Says:

thats the D’Alembert principle!

D’Alembert’s principle ? Oh, I am kind of getting now the way you are thinking about this. But, I’ll rather keep D’Alembert out of this- I am afraid we will really go far from this topic, if we start talking about analytical mechanics !

does inertial and non-inertial frames mean stationary and moving frames of reference? like a frame stuck to a billboard as opposed to one stuck to a moving car?

Oh!no! sorry ! I had taken for granted the definitions of frames(inertial and non-inertial) were wellknown – my mistake !

You know,I am resisting very hard my temptation to putforth the standard mathematical definitions of these objects. The advantage of math is of course that things can be defined very precisely and unambiguously.But, the disadvantage is that quite some mathematical agility is required to relate new math to the old already learnt physics.

Ok, let me give it a try ! By a frame I mean basically a set of observers who observe everything- events happening always and everywhere. These observers are by convention at rest with respect to each other.(Further, I prefer giving to each observer a set of three spatial vectors of unit length orthogonal to each other-e_x,e_y and e_z and again by convention all e_x’s are parallel to each other , all e_y’s are parallel to each other etc.)

If this frame is such that (according to the conventions of rest/spatial parallelism of the frame) there are no pseudoforces acting on any particle, such a frame is called a (global)inertial frame. As I’ve already mentioned in the article, if a spacetime does not allow such a frame, then the spacetime is said to be curved. A non-inertial frame is just a frame which is not an inertial frame.

8. nayagam Says:

then what is the change in geometry due to this pseudo force? I can see that the world appear to behave (move) differently when we go from a stationary to a moving frame of reference. is that it?

Almost, but not quite. Notice that, if you go from one frame to a second frame which is moving with a uniform velocity relative to first,you won’t see any weird thing. In fact, the principle of relativity demands that you SHOULD NOT see anything weird.

But, in the case of expanding frames, something weird does happen according to that frame’s observers. Those observers think that they are at rest with respect to each other,but, still, the distances between them change ! They are forced to blame the dynamics of space itself to explain why distances between them increase. Further, as I was telling in the post they are forced to blame the pseudoforces for this dynamics of space.

9. bharath Says:

got the inertial and non-inertial stuff. I think its coming clear to me now. thanks!

I knew only two kinds of mechanics. one that worked out through force interactions of Newton, the other through invariance principle of Lagrange and Hamilton. mmm.

a frame of reference is just a origin paiared with its coordinate system, right? I think I get that ok.

10. nayagam Says:

when you mean an expanding frame, you attach a distance measure to the frame which changes as the frame expands, so that two points are always at a contant distance with respect to the frame. but for an observer on the bill-board (say) the points will be receeding from each other? if this understanding is right, then why is it natural (any intuitive reasons?) to choose a distance measure depending on the frame of reference.

Hmm, well, there is only one distant-measure/metric for the whole spacetime.Only the components of the metric are different in different frames. But, hey wait, I thought I didn’t mention metric at all in this post ! Strike that out ! Actually I don’t follow your line of thought here. May be you want me to explain how metric fits into all this…

so the expading frame is like rubber like frame thats getting stetched due to pseudo forces (’invisible forces’)

well, I’m still not sure I am conveying to you what I want to convey.Let me give another try. The observers are radially expanding as seen from the inertial frame.They think they are at rest relative to each other, but they see that distances between them is increasing.According to inertial observer, this is just plain motion away from each other- No big deal! But, since the non-inertial observers are adamant on the fact that they are at rest and not moving, the non-inertial observers are FORCED to blame whatever they are seeing on the dynamics of space-time due to pseudoforces. that is briefly my argument. And by the way, pseudoforces are not anymore invisible than ordinary forces. What is invisible (in Newton’s mechanics)is the thing which produces these pseudoforces(the so called absolute space).

okie. so what is spacetime? As I understood it, the frame of referece only changes the way space gets represented. time appears to be a common denominator with no contraint on it.

Well, We owe our concept of spacetime to Minkowski. The main idea is that time is like one more dimension. I wrote a post some time ago
A time-like what ? which you might like to see along with 2-3 posts following it. There I am trying to show how relativity naturally leads to the concept of a spacetime.

11. nayagam Says:

a frame of reference is just a origin paiared with its coordinate system, right? I think I get that ok.

I rather think of frame as a set of orthonormal basis vectors at each event.What you are calling as a “frame” is more usually called a “co-ordinate system” or better “a co-ordinate chart” over some domain.

These two concepts can often be related. For examle, consider a two dimensional being on a sphere. Your “frame” would mean establishing a \theta,\ph” co-ordinates all-over the sphere(except possibly the poles). My “Frame” would mean the two unit vectors e_theta,e_phi at each point (except poles).