Minkowski Space Basics

1. Aug 7, 2010

kaleidoscope

Why is the vertical axis time multiplied by the speed of ligth? And, what is the purpose of using Minkowski space?

2. Aug 7, 2010

bcrowell

Staff Emeritus
Minkowski space, and the Lorentz transformations, tell us how space and time look different to different observers.

The Lorentz transformations mix x and t in a way that's loosely analogous to the way a rotation in the x-y plane mixes x and y. The existence of rotations shows that it would be silly to measure x in meters and y in feet. Similarly, the existence of Lorentz transformations shows that it would be silly to measure x in meters and t in seconds.

The c in relativity is not really the speed of light: --

FAQ: Is the c in relativity the speed of light?

Not really. The modern way of looking at this is that c is the maximum speed of cause and effect. Einstein originally worked out special relativity from a set of postulates that assumed a constant speed of light, but from a modern point of view that isn't the most logical foundation, because light is just one particular classical field -- it just happened to be the only classical field theory that was known at the time. For derivations of the Lorentz transformation that don't take a constant c as a postulate, see, e.g., Morin or Rindler.

One way of seeing that it's not fundamentally right to think of relativity's c as the speed of light is that we don't even know for sure that light travels at c. We used to think that neutrinos traveled at c, but then we found out that they had nonvanishing rest masses, so they must travel at less than c. The same could happen with the photon; see Lakes (1998).

Morin, Introduction to Classical Mechanics, Cambridge, 1st ed., 2008

Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51

R.S. Lakes, "Experimental limits on the photon mass and cosmic magnetic vector potential", Physical Review Letters 80 (1998) 1826, http://silver.neep.wisc.edu/~lakes/mu.html

3. Aug 7, 2010

Fredrik

Staff Emeritus
If you're asking because you've seen a diagram with x and ct on the axes, and are wondering why they put ct there instead of t, the answer is just that this is a convenient choice. It's convenient because it ensures that the world line of a light ray has slope 1 in the diagram. An alternative, which I prefer, is to take the axes to be x and t, and choose units such that c=1. This convention gives us the same diagrams, the same physics, and easier calculations.

• It makes it easier to state the theory.
• It makes it easier to develop an intuition about how things behave at relativistic speeds.
• It enables us to quickly find the solutions to many problems just by drawing a few lines on a piece of paper, instead of using algebra.