What is the confusion surrounding time as a coordinate in general relativity?

[AHSAN MUJTABA]

TL;DR

I want to know that how Einstein came towards the idea of considering time as a coordinate in his theory.

We study metrics, in them, we take time as a coordinate. I mean to say that if time is a coordinate then in normal mathematical language, we can have negative coordinate values as well. This confuses me a lot as I want to see and understand the concept from the true physicist's perspective. Please help me understand it. I know some of you may find it funny, but I really need to clear myself up.

 

[PeroK]

Time has always been a coordinate. And always taken positive and negative values. The SUVAT equation, for example: $$x = x_0 + ut + \frac 1 2 at^2$$ is valid for any value of $t$. And you can plot a position against time graph, with $x$ and $t$ as the coordinates.

 

[Ibix]

Maxwell's equations cannot describe a stationary light wave. In fact, they can only describe a light wave traveling at $c$. Einstein decided to consider the implications if it were true that light always travels at $c$ for all inertial frames, rather than trying to modify the equations to not say something so strange. He found that he had derived the Lorentz transforms, which Lorentz had already discovered as an ad hoc fix for Maxwell's equations without realising the implications. But Einstein's new perspective allowed him to realize that the Lorentz transforms were fundamental to mechanics as well, not just a quick fix for electromagnetism.

It was actually Minkowski, not Einstein, who pointed out that the Lorentz transforms had the form of a coordinate transform on a 4d non-Euclidean space, now called Minkowski spacetime. That was the insight that made time and space two parts of a whole, rather than two distinct phenomena. Einstein initially resisted the idea, I believe, but eventually ran with it, adding curvature to explain gravity.

Negative time is totally unexciting. If I have a stopwatch and start it now, I would describe one second ago as $t=-1$. As @PeroK says, this works perfectly well in non-relativistic physics too.

 

[sophiecentaur]

Negative time is totally unexciting.

It has disturbed and excited a lot of people so far. Negative t is ok in the Algebra but that 'ratchet' can be hard to deal with in the mind if you can't just 'accept' it.

 

[PeroK]

Is negative time a thing of the past?

 

[Ibix]

It has disturbed and excited a lot of people so far. Negative t is ok in the Algebra but that 'ratchet' can be hard to deal with in the mind if you can't just 'accept' it.

To expand a bit on my previous comment, then, in the context of coordinates negative time is just "the time before the time I called zero". It isn't really any more mysterious than representing latitude north and south of the equator as positive and negative numbers of degrees. In such a system one hemisphere might be called "negative space" by analogy, but there's nothing peculiar about it.

 

[ergospherical]

Negative time: no sweat. Imaginary time: panik...

 

[anuttarasammyak]

B.C. 100 = A.D. -100

 

[Dale]

Summary:: I want to know that how Einstein came towards the idea of considering time as a coordinate in his theory.

I mean to say that if time is a coordinate then in normal mathematical language, we can have negative coordinate values as well. This confuses me a lot as I want to see and understand the concept from the true physicist's perspective.

As others have said, negative time is fine. Consider a rocket launch. Famously they count down the time to the launch saying something like “t minus 10 seconds and counting”. They are at that moment at a coordinate time of negative 10 seconds. There is nothing wrong with that nor any physical difficulty.

 

[sophiecentaur]

Come on guys. We're being just a tad disingenuous here. What upsets us is the fact that we can choose to go in either direction on the x, y or z axes but we cannot experience the process of 'going in the negative time direction. Messing with signs in Maths is not an explanation of what's going on. The analogy / model of Maths falls down when used as we'd like to use it.
Is there an equivalent to the thermodynamic argument about entropy when we talk about the 'other' dimensions?

 

[ergospherical]

we cannot experience the process of 'going in the negative time direction.

You can, for example by writing $t' = -t$.

 

[Dale]

We're being just a tad disingenuous here. What upsets us is the fact that we can choose to go in either direction on the x, y or z axes but we cannot experience the process of 'going in the negative time direction. Messing with signs in Maths is not an explanation of what's going on. The analogy / model of Maths falls down when used as we'd like to use it.

I disagree. What you are pointing out is a separate issue than what the OP raised.

The inability to turn around in time has nothing to do with negative coordinate time as the OP asked about. It has to do with the fact that there is only one dimension of time.

 

[Ibix]

Come on guys. We're being just a tad disingenuous here.

I suspect that the OP might, indeed, mean something like "why can't we go backwards in time if time is part of spacetime". (The short answer is that there is no way to draw a future directed timelike worldline that turns into a past directed one without it being null or discontinuous somewhere, for the reason @Dale stated.) But that isn't what was asked, and I'd like to see if the OP says that's what was meant before I go off into too much of a tangent.

 

[sophiecentaur]

You can, for example by writing $t' = -t$.

What you are pointing out is a separate issue than what the OP raised.

The clue I got from the OP was the way he distinguishes between Maths and the Physicist. Perhaps the OP is suggesting or asking if there should be more acknowledgment of the essential difference between t and the other dimensions.
I guess the answer to the OP is that there is a difference but that people can't express it except to say you can't travel back in time.

 

[Ibix]

I guess the answer to the OP is that there is a difference but that people can't express it except to say you can't travel back in time.

No, we can express it clearly. There's only one timelike dimension, which has the consequence that you can't reverse direction in that dimension without exceeding the speed of light, which you can't do. I'm just not yet sure if that's what we're being asked.

 

[Dale]

I guess the answer to the OP is that there is a difference but that people can't express it except to say you can't travel back in time.

I agree with @Ibix, it can certainly be expressed easily enough, but I don’t see that in his question.

The differences between time and the other dimensions are the sign in the metric and the fact that there are three dimensions of space and only one of time. The sign of the metric physically means that timelike intervals are measured by clocks while spacelike intervals are measured by rulers. The presence of multiple spatial dimensions allows closed spacelike curves. The presence of only a single temporal dimension prevents closed timelike curves.

 

[George Jones]

I see nothing in the original post that excludes general relativity, and the thread's title includes "General Relativity". In which case, all of the statements below are false.

The inability to turn around in time has nothing to do with negative coordinate time as the OP asked about. It has to do with the fact that there is only one dimension of time.

(The short answer is that there is no way to draw a future directed timelike worldline that turns into a past directed one without it being null or discontinuous somewhere, for the reason @Dale stated.

The presence of multiple spatial dimensions allows closed spacelike curves. The presence of only a single temporal dimension prevents closed timelike curves.

The situation is much more subtle than this. If we are restricting to the spacetime of standard special relativity that is both flat and topologically $\mathbb{R}^4$, then this statement is true; otherwise, this statement is false.

If either of these condition is relaxed, it is possible to have one dimension of time and closed timelike curves. Examples 1) flat spacetime with topology $S \times \mathbb{R}^3$; 2) Godel's spactime which is curved, and which has topology $\mathbb{R}^4$.

 

[DaveC426913]

B.C. 100 = A.D. -100

< pedantry >
B.C. 100 = A.D. -101

There is no year zero.
< /pedantry >

 

[Dale]

I see nothing in the original post that excludes general relativity, and the thread's title includes "General Relativity". In which case, all of the statements below are false.

The situation is much more subtle than this. If we are restricting to the spacetime of standard special relativity that is both flat and topologically $\mathbb{R}^4$, then this statement is true; otherwise, this statement is false.

If either of these condition is relaxed, it is possible to have one dimension of time and closed timelike curves. Examples 1) flat spacetime with topology $S \times \mathbb{R}^3$; 2) Godel's spactime which is curved, and which has topology $\mathbb{R}^4$.

Good point, but in those cases the false part is “The inability to turn around in time”. The explanation that follows is correct for any spacetimes that actually have that inability regardless of curvature or topology. The restriction to flatness and R4 is too strong.

 

[DrGreg]

< pedantry >
B.C. 100 = A.D. -101
< /pedantry >

< pedantry >² Isn't it BC 100 = AD –99 ? < /pedantry >²

 

[DaveC426913]

< pedantry >² Isn't it BC 100 = AD –99 ? < /pedantry >²

You are correct. And I am wearing egg.

BC 100, ... BC 3, BC 2, BC 1, AD 1, AD 2, AD 3, ... AD100

So:
BC n is equivalent to AD -(n-1)
eg:
n=3: BC 3 = AD -(3-1) = AD -2
n=100: BC 100 = AD -(100-1) = AD -99

 

[anuttarasammyak]

It is a peculiar mathematics of no zero. Time interval of years for the same date, e.g. 4 July, is given as
A.D.m-B.C.n=m+n-1=m-(-n+1)=n-(-m+1)
In order to satisfy it
A.D.m=B.C.(-m+1)
B.C.n=A.D.(-n+1)
But it brings prohibited A.D.0 and B.C.0. So in contrary to my previous post no minus sign should be introduced to A.D. and B.C..

 

[sophiecentaur]

< pedantry >² Isn't it BC 100 = AD –99 ? < /pedantry >²

I remember a similar awkwardness in using a binary scale with Magnitude and Sign when dealing with ADC data. You either get a jump from +0001 to -0001 and a hiccup at a non existent zero or you have to have +0 and -0 (two concident values). The hiccup can be avoided by using a half (least significant) bit offset and you step over zero with a +1/2 and -1/2 and get a smooth curve over the origin. (Same as with crossover distortion in push-pull audio amps.)
So perhaps the date problem could be overcome with a six month offset and produce a symmetry with AD-1/2 and BC+1/2 and a non-existent zero.

 

[DrGreg]

I remember a similar awkwardness in using a binary scale with Magnitude and Sign when dealing with ADC data. You either get a jump from +0001 to -0001 and a hiccup at a non existent zero or you have to have +0 and -0 (two concident values).

I seem to remember a similar issue with the way Microsoft represented dates (and probably still does). A date-and-time was represented as number of days since a reference date (31 Dec 1899 I can't remember if it was 1 Jan 1901 or 1 Jan 1899), with fractional values representing the time of day, but values between 0 and 1 represented a time with no date, leaving a discontinuity in the date-and-time scheme between values < 0 and values ≥ 1.