- #1

- 185

- 1

sqrt(x^2 + y^2)

and in 3 dimensions it's

sqrt(x^2 + y^2 + z^2)

but in 4 dimensions, with the 4th being the change in time, why is it

sqrt(x^2 + y^2 + z^2 - (ct)^2) ?

Can somebody please explain?

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- Thread starter nhmllr
- Start date

- #1

- 185

- 1

sqrt(x^2 + y^2)

and in 3 dimensions it's

sqrt(x^2 + y^2 + z^2)

but in 4 dimensions, with the 4th being the change in time, why is it

sqrt(x^2 + y^2 + z^2 - (ct)^2) ?

Can somebody please explain?

- #2

Nabeshin

Science Advisor

- 2,207

- 16

If you recall from elementary discussions of the line element, ds^2, what we're really trying to identify is some invariant quantity -- that is, something all inertial observers will agree on. The simple fact is that if you do not use a negative sign, it is not an invariant quantity.

More mathematically speaking, the negative sign comes from the fact that relativity is limited to special sub-class of all riemannian manifolds known as Lorentzian manifolds (those that have - + ++ signature). The language of general relativity (and consequently, special relativity), is formulated in terms of these Lorentzian manifolds which make a distinction between null, timelike, and spacelike events.

- #3

- 422

- 4

- #4

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- 0

- #5

K^2

Science Advisor

- 2,469

- 29

Except, it's not that straight forward. If you have two reference frames that are moving relative to each other, time coordinate of one has projection onto spacial coordinates of the other, and in both, you'll have exactly the same negative sign in the metric. So it isn't something unique about a particular direction in space-time. It's something about the structure of space-time that gives you an ability to pick one of infinite number of possible directions to be different than the rest of the directions you are going to pick.Ah, yes! This is perhaps the most important negative sign in all relativity! If it were just normal (ct)^2, then it would be identical to 4-dimensional Euclidean space. I guess it boils down to the fact that time is NOT just another spatial dimension -- there is something unique about it.

And yeah, this doesn't pack very well in a brain that's used to Eucledian space. I've seen pure mathematicians get downright upset when I suggested that a metric does not have to be positive.

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