Question

How are time and space linked?

Please provide a long answer to the question(s) above, using diagram and formula whenever possible.

Answer 1

Mathematically, and in accordance with relativity, they are in some sense interchangeable, but we do know that they form co-equal parts of a larger 'thing' called space-time, and it is only within space-time that the most complete understanding of the motion and properties of natural objects and phenomena can be rigorously understood by physicists. Space and time are to space-time what arms and legs are to humans. In some sense they are interchangeable, but you cannot understand 10,000 years of human history without including both arms and legs as part of the basic human condition.

Space and time are independent in Newtonian physics. The spatial distance between two events is relative to the observer but the time elapsed between them is absolute and same for every Newtonian observer.

Let me give you an example. Suppose, you ("A") are in a train which is moving with respect to an inertial observer (say an observer "B" standing on the platform) with a constant velocity. Since both of you are inertial observers, by Galilean principle of relativity, you can consider yourself to be at rest and "B" to be moving at the opposite direction with constant velocity.

Now suppose you throw a ball vertically upward. You will see that it would land on the same place after some time, say, after 02 seconds, according to your clock. Since, the event of throwing the ball and landing the ball are at the same position, the distance between these two events is zero. From the point of view of "B" also, the time taken between the same two events is exactly the same i.e. 02 seconds (because "time" in Newtonian mechanics is absolute) but the distance is not zero. This is because, within this period, the train has moved some distance with respect to "B".

So you and "B" will always agree about the time taken between the two events but you can't agree about the distance between the events. In other words, in Newtonian physics, space is not absolute but time is absolute.

Also, in Newtonian physics, space and time are deliberately undefined concepts. This is because, in Newtonian world view, space and time provide the background where physical theories are to be built. Space and time have to be kept completely pure from other concepts. If we define space and time in terms of some properties of the world then they can't provide a faithful background for description of physical phenomena. This was Newtonian philosophy.

It is noteworthy, that this strategy was attacked most vocally by Gottfried Leibniz, according to whom, the world is a clock rather than world has a clock. For Leibniz everything was to be defined in terms of relations and no concept can stand on its own.
Whatever, Newtonian strategy was spectacularly successful and it worked. That's all that mattered. But until electrodynamics came. Classical Electrodynamics confronted Newtonian conception of space and time because electrodynamics predicted a preferred velocity, the velocity of light, which in turn appeared to imply a special, privileged frame of reference. That is in direct contradiction with classical, Galilean, principle of relativity which states that all inertial frames of references are equivalent in the description of the world and there is no way, an inertial observer can claim to be in motion absolutely by exploiting any law of nature.

There were several heroic attempts to solve the puzzle in terms of modifying the laws of electrodynamics, in a way to make it compatible with classical relativity but all in vein. It might sound crazy but some physicists were indeed ready to accept the bizarre suggestion that principle of relativity might be valid for mechanics but not for electrodynamics!

Then came the patent clerk, Albert Einstein, with his revolutionary proposal. (We must not forget that Poincare was also very close to the final solution. But Einstein's arguments were closer to physics and clarified the matter in much more clear light.)

Einstein argued that the problem can be solved in one stroke if we are willing to reject Newtonian concept of absolute time. Each inertial observer has his own personal time and there is no such thing as an universal absolute time that every inertial observer can agree on.

Einstein made the following two claims:

1) The laws of physics (be it mechanics or electrodynamics) are same for all inertial observers. The principle of relativity is valid and beyond doubt.

2) The speed of light in vacuum is same for all inertial observers.

The second claim clearly implies that the concept of time in Newtonian physics should be modified otherwise two observers in relative motion can't agree on the speed of light. Two events, simultaneous for one observer can be non-simultaneous for another.

With these two assumptions, Einstein derived the correct transformations of space and time for two inertial observers. They are called Lorentz transformations because Lorentz proposed those equations to explain the empirical observations of the Michelson Morley experimental results. Einstein deduced them from elementary assumptions.

According to Lorentz transformations, one observer's space can be another observer's space and time and vice-versa. This means space and time are intertwined and can not be treated separately. When describing physical phenomena, space and time should be combined into a four dimensional mathematical space, called spacetime. The points in this space are possible events.

It is undeniable that Einstein was positivistically motivated while proposing all these ideas. He was little more closer to Leibniz than Newton as far as the role of space and time is concerned. But ultimately what matters is what works and not philosophical mumbo-jambo.

But space and time are not exactly same in relativity. The metric signature is (-,+,+,+) or (+,-,-,-) as is evident from the following:

ds²=dx²+dy²+dz²−c²dt²
From this we can infer which portion of spacetime can be causally related with which.

If two events are such that ds2>0ds2>0 then they are causally unrelated and can't influence one another. We call the events are spacelike separated. We can not say which event is earlier since there is no causal connection.

If ds2<0ds2<0 then the events are timelike separated and one can influence the other and the first one is called the past event and the second one is in its absolute future. One can't construct a spacetime where the time order of these two events can be reversed.

There is obviously a third possibility that ds2=0ds2=0. We call it lightlike separated which should not concern us right now.

But there is still the question of gravity, which according to Newton, is an action at a distance interaction. But for an action at a distance force "time" has no place in its formulation. It means gravitational influences should travel with infinite speed. But that is in contradiction with special relativity.

How to solve the problem? Einstein started pondering about the problem right after 1905. He struggled and struggled. Then in 1907, while still at the patent office, he got his brilliant insight which would be known as the principle of equivalence. Einstein at once became convinced that the basic problem had been solved. All that remained was the correct formulation of the idea. This formulation would take him to almost 08 years of long, painstaking journey with repeated failures, right avenue, then wrong avenue, then again the right with alternating confidence on the correct path and then finally the correct equations with its full glory. An amazing intellectual adventure of a lone man.

The final product is called the general theory of relativity. Compare to general relativity, special relativity was a child's problem (Einstein's own words). General relativity went far ahead than special relativistic idea of space and time. Far from providing an inert background, spacetime in general relativity, is a dynamic entity which affect everything that happens in the universe and is affected. Einstein's equations are highly nonlinear and are extremely difficult to solve. This nonlinearity is superbly expressed by John Wheeler, "spacetime tells matter how to move; matter tells spacetime how to curve".

What does it mean, when we say a space is curved? It simply means the metric tensor is no longer Euclidean. The special relativistic metric tensor is given as

Because of the negative sign at the g11g11 the spacetime in special relativity is pseudo Euclidean.

A typical metric in a highly curved spacetime is like the following:

When gravity is absent, the spacetime becomes flat and the metric becomes as given in the first figure and general relativity reduces to special relativity.

What does quantum mechanics has to say about space and time? Nobody knows the final words but as things stand today, in quantum field theory (in flat spacetime), there is a background spacetime having the same structure as in special relativity. However, one can do QFT in curved spacetime also. Anyways, quantum theory by itself doesn't modify or dictate the structure of spacetime.

But in quantum gravity, this may not be so. There are different possible ways about how quantum theory can affect the structure of spacetime. This is a domain of active research. One of the most radical discovery within string theory, in recent years is Ads/CFT correspondence which is already partially proved. The correspondence shows that two different theories with different spacetime dimensions can describe the same physics! Why is this so deep and revolutionary? Because firstly, it's kind of giving hints that spacetime may not be fundamental and secondly physics with gravitation can be replaced with physics without gravitation in a dual picture!
There are various other implications and possibilities yet to be realized. Again nobody knows the final picture. Perhaps we are still just kids with our diapers. Who knows?