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.