Time-Like Intervals: Can't Find an Inertial Frame for Events?

[Master J]

For a time-like interval between 2 events, it is impossible to find an inertial reference frame in which the events occur at the same time. This can be seen from the space-time interval s$^{2}$=c$^{2}$t$^{2}$-l$^{2}$ where s must be real number for a time-like interval.

However, how does it follow from this that one cannot find a frame in which they occur in the reverse order?

 

[PeterDonis]

Inertial reference frames have to be related by a Lorentz transformation, and a Lorentz transformation can't change the sign of the time component of the interval.

 

[PAllen]

For a time-like interval between 2 events, it is impossible to find an inertial reference frame in which the events occur at the same time. This can be seen from the space-time interval s$^{2}$=c$^{2}$t$^{2}$-l$^{2}$ where s must be real number for a time-like interval.

However, how does it follow from this that one cannot find a frame in which they occur in the reverse order?

You can't find such a frame corresponding to any valid state of motion relative to the given events, that changes their order. This follows from the Lorentz transform - time is never reversed for any relative speed v. Further, no valid coordinate transform can change timelike to spacelike (this follows simply because all invariants are preserved under coordinate transform, esp. timelike interval).

However, you can do a global coordinate transform that reverses time. This effectively just 'runs the universe backwards'. All classical interaction laws are time symmetric. While this will be locally valid, in any realistic physical system, one of these time directions will match growth of entropy, while the other won't. We say that that the direction of entropy growth is the direction time is experienced.

 

[Master J]

But how does the sign of the time component matter, since time squared is in the interval?

 

[Master J]

Ah, so what I am gathering here is that for an event A with time component t, and another event B say at t=0, one cannot transform to another frame (with a Lorentz transformation) so that A is now at -t since the Lorentz transform cannot change the sign of t.

That makes sense! Cheers guys!

 

[PAllen]

But how does the sign of the time component matter, since time squared is in the interval?

I thought both answers you've received explained that. Put any velocity you want, positive or negative, into the Lorentz transform, and it doesn't change time direction.

I also noted that you can globally reverse time coordinate, but that doesn't really change physics - you can tell you are looking at the universe run backwards (people will be expelled feet first from pools, to arc onto diving boards, etc.).