Today’s post revisits one of my favourite themes in Theoretical Physics – time. In previous posts we’ve looked at the idea of the present and how the theory of Special Relativity refutes its absolute existence – and we’ll be drawing on this idea again here. Today I want to discuss the idea of ‘proper time’ and tell you what it is, alongside a deeper explanation of the geometry of spacetime.
Back in the day it was believed by many, including the great Newton who put together the three laws of motion, that time was absolute. This meant that wherever you went in the universe, a clock at one end of the cosmos would be ticking in synchronisation with one at the other. As though, there was a universal time, running on an almighty clock in the sky and all other clocks in the cosmos would adhere to its pace. Not so. With the discover of Special Relativity, we learn that the ticking of one’s clock and the time measured between events, depends on where you are in the universe. It depends on whether you’re near a gravitational field and how fast you are moving. The closer you move to the speed of light, the slower your clock will tick. The closer you are to a heavy mass, the slower your clock will tick. Let me just caveat here – this is not a special property of just clocks! The is the property of time as we know it, itself. A clock is just a mechanical object that we have devised, which has a constant periodic tick, from which we can then measure other events against. It is not just this tick that slows down, it is all atomic processes. The beating of your very heart would slow down in a strong gravitational potential or when travelling at a fast speed – for in essence your heart (if regular in its beating I hope!) is nothing more than a type of clock. If you need a reminder of how time is warped by relativistic effects I refer you to these two posts ‘Does the present really exist‘ and ‘Black Holes: #1 Falling In‘.
From now i’ll take it as understood that we all accept that modern physics tells us that time as we know it is not absolute – (accepting this deep down in your being is another matter entirely and I’m sure even the greatest theoretical physicists have trouble with this).
When describing an object we use 4 coordinates, three for the spatial position of the object and one for the time: x, y, z, t. Now the framework used to describe the geometry of everyday space is called, by mathematicians, Euclidean geometry. Another very useful tool employed to visualise spatial geometry is a standard two-dimensional graph. If we plot on a page two axes – y and x, one representing one direction e.g. North and the other a direction perpendicular to it e.g South and then plot the relative positions of two objects on this graph, the line joining them represents the distance between these two objects.
Of course in real life there are three spatial dimensions, so we can also plot a three dimensional graph. Imagine a room. The z axis protrudes out the floor straight up to the ceiling, and the x and y axis exist in the floor at right angles. Your feet exist at one point in this graph and the door knob at another, the straight line across these three dimensions represents the distance between your feet and the door. Simple right? I’m sure you knew that already. Where am I going with this? Bear with me.
(A 2D representation of a 3D graph!)
Next step. How do we describe the geometry of four dimensions? Well this is the geometry of space-time. The graph we use here is called a space-time graph or space-time diagram. Now unfortunately I can’t create a nice visual analogy for you this time, because if you can visualise four-dimensions you’re a super-human (or an alien). We have the three-dimensional graph from before with an extra axis, protruding out in an extra fourth dimension representing time. Now again we plot two things into the graph – we call these things events instead of objects now because they include time. For example, Event A could be Big Ben chiming at 12pm on the 18th June 2017 and Event B could be New Year’s Eve 2017 at the Eiffel Tower. Now what does the line on the space-time graph connecting these two events signify? Clearly not distance. The line connecting two events on a space-time four-dimensional graph represents a quantity theoretical physicists called proper time.
Proper time?! What on earth is that I hear you say. Let me explain. Proper time is a construct made from the difference in the four coordinates x, y, z, t between two events (along with a factor of c – the speed of light). And whereas the time (t) between two events is not always measured to be the same because of relativistic effects (somebody running fast will measure the time between two events to be less than somebody standing still) any observer, regardless of their place in the universe or their speed will always measure the proper time (the combination of the change in all x, y, z and t) to be the same. This is very profound indeed – we’ve found a quantity, composed of the spatial and temporal coordinates that is invariant. The symbol for proper time is tau, τ. The equation is given here:
Why do physicists call it proper time? Well my inkling is because, as I mentioned before, we as humans are so deeply unsettled by the fact that time is not measured the same for all that we wanted to give the quantity that we did find to be fixed, a connection to our beloved time.
So just to re-iterate the key points here. A space-time graph, is a graph of four dimensions which is exists in four-dimensional space-time geometry (x, y, z and t). Two events are plotted on the graph and the magnitude of the line joining them represents the proper time between the two events. This quantity is measured to be the same for any observer, regardless of their motion. So as not to confuse these special 4D line with 2D or 3D lines in normal Euclidian geometry, we call the lines in four-dimensions worldlines.
(A 2D representation of a 4D space-time graph – even worse! You can ignore the world sheet and world volume for today.)
In a way this discovery does alleviate my unsettled feelings to do with the relative nature of time. With proper time there now does exist a quantity, composed of all four dimensions whose measurement remains invariant between any two events in this universe. The invariance, we initially believed held for time, exists instead on a higher dimensional level, incorporating time as a component. And thankfully it does, for a universe where everything was truly relative and nothing absolute wouldn’t sit well with our burning desire to seek out an underlying simplicity to nature.