Space-Time Metrics

Personally created image

The Klingon Canoe

An analogy was made in Taylor and Wheeler’s book “Exploring Black Holes” that I just love. It involves an overturned rowboat. I took some personal liberties with this one, but feel the heart of the analogy remains.

Let’s take a canoe, made by the Klingons from Star Trek, so it has an invisibility cloak. How do you measure it’s shape? Below is an image of the canoe, if one could see through it’s cloaking device.

Image courtesy of "Exploring Black Holes".
Image courtesy of “Exploring Black Holes”.

One clever way is to drive a nail in at certain intervals along it’s surface (let us pretend this act doesn’t compromise it’s invisibility cloak).

Image courtesy of "Exploring Black Holes".
Image courtesy of “Exploring Black Holes”.

Now we can attach strings to each nail, strings of known distance, and get a pretty good idea of the shape of the canoe.

Inviso Canoe 3
Image courtesy of “Exploring Black Holes”.

So, to someone that can’t see through the invisibility cloak, this is what we have. A representation of strings and nails to give an idea of the shape of the canoe.

Invisible Klingon canoes…what does this have to do with Space-Time structure?


Space-Time Metrics

Image courtesy of http://plato.stanford.edu/entries/spacetime-holearg/

What is a metric? True, it is another name for the S.I. units (meters, kilometers, kilograms) used in modern science. But in this case, it has another meaning. According to Taylor and Wheeler, “a metric provides the means by which we meter or measure spacetime”. The above analogy was to get the reader used to the idea of the nails. But in physics, we can’t go around driving nails into Space-Time. Could you imagine such a thing? Nails driven into empty space?!

But, we can do one better. Events. Events are the nails in physics. Remember, observers in different reference frames may not agree on the occurrence of an event. Metrics help us to deal with this. Physics teachers love to discuss events as firecracker explosions. If we can measure the time and spatial separations between events, we can get an idea of the underlying structure of the Space-Time in which they occurred.

There are two types of Space-Time metrics: those for flat space, and those for curved. I will cover metrics for flat space-time here, the metric for curved space-time is on the next page.

Flat Space Metrics

We refer to “flat” space-time as areas where there are no massive bodies. In areas like this, we can refer to two types of metrics: time-like and space-like.

Time-like:

Image courtesy of "Exploring Black Holes"
Image courtesy of “Exploring Black Holes”

The time-like metric is for events that have a great time separation than spatial. The “τ’ in the equation is referred to as the “wristwatch time” between the two events.

 

 

Space-like:

Image courtesy of "Exploring Black Holes".
Image courtesy of “Exploring Black Holes”.

The space-like metric is used when the spatial separation between events is greater than their time separation. The “σ” represents the spatial separations between the events.

Of course, both statements are very simplified. Rarely do events occur in a plane that allows for only x and y spatial considerations. Often you will see these statements expressed in terms of differentials, and in different coordinate systems.


WHAT ABOUT SOLUTIONS TO EINSTEIN’S EQUATION IN CURVED SPACE, AROUND A MASSIVE OBJECT? IN WALKS KARL SCHWARZSCHILD…

The Schwarzschild Solution