## Wednesday, December 31, 2008

### The Counterintuitive Time: 2. The Geometric Time

This is a series of posts about the counterintuitive nature of time in Physics. This post tries to identify the problem in accepting the geometric nature of time implied by Special Relativity, as well as the differences between space and time in relativistic spacetime.

In Newtonian Mechanics, the world evolves deterministically. The time is a parameter similar to the space coordinates. The PDE describing the evolution respect symmetries at orthogonal transformations of space, and at time translation and time reversal. Another symmetry is related to the speed of the inertial reference frames: the laws do not depend on the speed of an inertial frame.

A challenge of the Galilean relativity is provided by the Maxwell’s equations. The Electrodynamics suggested another group of symmetries, the Poincaré group, and its Lorentz subgroup, which are associated to the Special Relativity.

The introduction of the Lorentz transformations shed a new light on the nature of time. The time is no longer a parameter, but it gains a geometric meaning, which brings new counterintuitive aspects. The geometric meaning of time comes from the Lorentz invariance. The Lorentz transforms can “mix” space and time dimensions, like a spatial rotation can mix two directions of space. The relativity of simultaneity, which is a central point of Einstein’s theory, provides a physical interpretation of this character. This challenges our intuition, because it suggests that spacetime is a single geometric and timeless entity. Each direction in the Minkowski spacetime corresponds to a speed. The relative speed between two such directions can be obtained by applying the hyperbolic arctangent to the angle between them. This shows that two inertial frames moving with a relative speed, have different time direction in spacetime.

When somebody hears about the Minkowski spacetime as a symmetric space, may think why we couldn’t move through time like we are moving through space. The usual answer involves the idea of lightcone, but this explanation is not enough. But let us first discuss the lightcone and the causal structure of Relativity.

The lightcone is the set of all spacetime directions which corresponds to light speed. The 4-vectors from inside the cone, represents time directions, and the ones from outside, spatial directions. The squared norm of a spacelike vector has opposite sign than the squared norm of a timelike vector, and the lightlike vectors have zero norm, being also named null vectors. The Lorentz transformations preserve the norms, therefore they cannot be used to turn a timelike vector into a spacelike vector.

It seems impossible for an object having a velocity smaller than the speed of light to change smoothly the direction in spacetime and go back in time. The main reason is that its velocity will need to become the speed of light, and then larger (to go out of the light cone). But what is the problem with a body being accelerated to the speed of light? The answer is that we would need an infinite amount of energy for doing this. When the body increases its speed, its mass also increases, and the energy required for increasing further its speed becomes larger. For going to the speed of light, we will need to give it an infinite amount of energy.

Although we understand that the Relativity explains well our limitations in moving through time like we are moving through space, this difference between space and time are still so deep rooted in our intuition, that we find very difficult to accept the geometric nature of time.

A second counterintuitive aspect is the difference between the spacelike and the timelike vectors. If they can be rotated one into another by Lorentz transforms, this rotation is only partial, because we cannot transform a spacelike vector into a timelike vector. This asymmetry is not that annoying from mathematical viewpoint, and, as we saw from the previous argument, it is even useful. But many find it disturbing, and they feel like there is a need to replace the Lorentz metric with a Euclidean one (by some mathematical trickery). In general, these attempts ended up by complicating the things, and the mainstream physicists remained with the Lorentz metric. But, I cannot deny that there may be persons who consider simpler the Euclidean approach, because the price of accepting an indefinite metric seems too high for them. Maybe it is a matter of taste.