Why Is Time Considered the 4th Dimension in Einstein's Theory?

[Xeinstein]

Why should time be considered as a 4th dimension?

 

[CaptainQuasar]

Look at [post=1602401]this thread[/post]. It's simply done for convenience of mathematical analysis. You could just as easily have temperature or magnetic field strength for a 4th dimension.

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[JesseM]

It's true that any variable can be treated as a dimension, but the reason it's so popular to think of spacetime in geometric terms in relativity goes beyond that. It also has to do with the fact that, although different frames can disagree on the spatial distance or the temporal separation between events, they will all agree the invariant "spacetime interval" between them, so this can be thought of as a kind of "distance" in 4D spacetime. In 3D space the spatial distance between points is given by the Pythagorean theorem d = \sqrt{x^2 + y^2 + z^2} (which should give the same answer regardless of how you orient your x-y-z axes, since all cartesian coordinate systems in Euclidean space agree on the distance between points), and the formula for the spacetime interval is similar, \sqrt{c^2 * t^2 - x^2 - y^2 - z^2} (which gives the same answer regardless of which frame's x,y,z,t coordinates you're using). Along the same lines, there is the fact that the "geodesics" in the curved spacetime of GR are the ones that locally have extremal values (usually minimal values) of proper time (which is what the spacetime interval is giving you in flat spacetime), they don't have extremal values of spatial distance.

And one other reason for thinking of spacetime in geometric terms is the relativity of simultaneity, which tells you that there is no physically preferred way to divide up 4D spacetime into a stack of 3D "instants" showing how the configuration of matter in space evolves over time.

 

[peter0302]

I'd like to point out that it's possible to extend Euclidean geometry to five dimensions, keep almost all of S.R., and maintain the Pythagorean theorem as-is, as discussed in this fascinating paper posted in Independent Research:

https://www.physicsforums.com/showthread.php?t=103977

That's an even more persuasive reason, IMO, to treat observer time as a 4th dimension.

 

[kdv]

Why should time be considered as a 4th dimension?

It's the fact that the speed of light is an invariant, a truly fundamental quantity. because of this, if you take any two events and you assign them coordinates (as measured in some specific frames), the quantity c^2 (\Delta t)^2 - (\Delta x)^2 - (\delta y)^2 - (\Delta z)^2 gives a number that will be the same as measured by any other observer (using the intervales measured in their own frames). This is what unifies time and space into a four-dimensional manifold.

 

[GearsofWar]

In Einstein's original papers, doesn't he say:

x4 = ict ?

It seems then that the fourth dimension is ict as x4 = ict.