Relativity *Does Not* Say Time And Space Are On The Same Footing.

Many people claim relativity says that space and time are on equal footing and since we can move back and forth in space it doesn't make sense that we can only move in one direction in time. (Admittedly I have in fun used the argument myself but now I would like to give the reason I think the argument is wrong.)

To do so, I am going to quote from David Tong's lecture notes on String Theory that explains why time is *not* on the same footing as space even in relativity, and you are constrained to move forward in time after all unlike space.

First, here is the argument in a nutshell:  We want our physics equations to be Lorentz invariant, and given Lorentz transformations mix up space and time, we want need equations where space and time are on the same footing.  However!!! To do so we have to add a fictitious degree of freedom that isn't real.  Therefore, this idea that space and time are on the same footing is just a mirage we created by adding a fictitious degree of freedom to get the equations to have a symmetry we want.  Furthermore, given this fictitious degree of freedom is unphysical, your equations produce non-physical answers until you remove this non-physicality using constraint equations. Once you do you find that you have always been constrained to move forward in time... just as common sense dictates.

Now to Tong:

Although the [non-Lorentz invariant] Lagrangian is correct, it’s not fully satisfactory. The reason is that time t and space x play very different roles in this Lagrangian. The position x is a dynamical degree of freedom. In contrast, time t is merely a parameter providing a label for the position. Yet Lorentz transformations are supposed to mix up t and x and such symmetries are not completely obvious in (1.1). 

So, even though a Lagrangian where space and time are not on the same footing is physically correct, in order to have the fancy symmetry of being Lorentz invariant we would like to change the situation.

In this course we will... promote time to a dynamical degree of freedom. At first glance, this may appear odd: the number of degrees of freedom is one of the crudest ways we have to characterize a system. We shouldn’t be able to add more degrees of freedom at will without fundamentally changing the system that we’re talking about. Another way of saying this is that the particle has the option to move in space, but it doesn’t have the option to move in time. It has to move in time. So we somehow need a way to promote time to a degree of freedom without it really being a true dynamical degree of freedom! How do we do this? The answer, as we will now show, is gauge symmetry. (I added emphasis)

Again, the first mentioned action describes reality and claims only space is a dynamical degree of freedom, not time.  We want our equation to become Lorentz invariant sure, but we can't just start adding degrees of freedom where they don't exist without paying the price of our equations not being physical:

[After adding Lorentz invariance] Naively it looks as if we now have D physical degrees of freedom rather than D−1 because, as promised, the time direction X0 ≡ t is among our dynamical variables: X0 = X0(τ). However, this is an illusion. To see why, we need to note that the action has a very important property: reparameterization invariance...The upshot of this is that not all D degrees of freedom X  are physical. For example, suppose you find a solution to this system, so that you know how X0 changes with τ and how X1 changes with τ and so on. Not all of that information is meaningful because τ itself is not meaningful.

And, since things are no longer physical, to bring reality back to our theory the unphysical aspects of our equations need to be constrained away:

The fact that one of the degrees of freedom is a fake also shows up if we look at the momenta. These momenta aren’t all independent. They satisfy p^2+m^2 = 0. This a constraint on the system... From the worldline perspective, it tells us that the particle isn’t allowed to sit still in Minkowski space: at the very least, it had better keep moving in a timelike direction with p0^2 >= m^2.

So, just as we perceive with our everyday senses: time is not on the same footing as space after all.  The unphysical aspect of our equations giving the mirage that this was the case needs to be constrained away once again giving results showing we are allowed to freely move in space, but time is forced upon you as always moving forward in accordance with p0^2 >= m^2.  (Here p is your momentum).

So in Conclusion, and to repeat:  Don't be fooled.  The "equations of relativity show time and space are on the same footing" claim is wrong.  The equations only appear that way because we have introduced a fictitious degree of freedom to accomodate Lorentz invariance.  In reality, that un-physicality needs to be constrained away to get physical solutions, and when they are, you quickly find you cannot move through time in any direction as you do through space but are constrained to move forward in one direction in accordance with p0^2 >= m^2.  This constraint is often called the "on-shell" condition for those who care.

So time always moves forward, even in relativity... just as common sense experience shows.