Geometry of Spacetime

You mean -i*t rather than -t since the coordinates are squared in the given equation.

The glib answer would be that the universe just does not work that way. This is not a calculus question per se, but a special relativity question.

http://www.iep.utm.edu/proper-t/#H1

quote:

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STR predicts that motion of a system through space is directly compensated by a decrease in real internal processes, or proper time rates. Thus, a clock will run fastest when it is stationary. If we move it about in space, its rate of internal processes will decrease, and it will run slower than an identical type of stationary clock. The relationship is precisely specified by the most profound equation of STR, usually called the metric equation (or line metric equation).

(1) [sorry I'm too lazy to latex the equation]

This applies to the trajectory of any physical system. The quantities involved are:

Dt is the amount of proper time elapsed between two points on the trajectory.Dt is the amount of real time elapsed between two points on the trajectory.
Dr is the amount of motion through space between two points on the trajectory.
c is the speed of light, and depends on the units we choose for space and time.

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What does it mean to be 'at rest'?

Thanks. I was really checking to see what Catholic Scientist meant by 'at rest'?

Taq has anticipated where my question is going. As I understand your answer, you seem to believe that there is some rest state that is impossible to achieve. Instead the situation is that some objects are at rest with respect to some frame, possibly an inertial frame. However there is no special inertial frame which defines motion at rest.

Edited by NoNukes, : No reason given.

This is true. However, there is a natural set of units that eliminates this problem. If the time and distance units are chosen so that the speed of light equals one, then we have eliminated the issue you describe above.

Assuming units where c=1, you can express the proper time as follows:

Where distances are changing at a constant rate with respect to time, then you can use this expression:

This is the form that I have seen used to show how the twin paradox works. By the way, just because an expression generates an error on your calculator does not mean that the expression is improper.

Edited by NoNukes, : No reason given.

You can develop the equations with either negative time contributions and positive space contributions or vice versa.

The "d"s in the first equation indicate that dt, dx, dy, dz are differential values. The integral is a line integral over the trajectory of a particle.

In the equation with the deltas, deltas mean 'change'. The second equation can be be used where the change in time, and x, y, z coordinates in a case where those quantities each vary linearly with time from a starting point to the ending point.

Your questions suggest that you've got quite a bit of studying including learning a tiny bit of math before you are going to understand special relativity. Try reading the wikipedia article I linked as a starting point.

Edited by NoNukes, : No reason given.

I have some patience, but to date you don't seem willing to make much of an effort on your own.

Let's discuss what the equations are mean. The equation for proper time gives the time that an observer traveling a trajectory would measure. Observers not following that same trajectory could measure different times. This is explained in a fairly detailed manner with a couple of example calculations at the Wikipedia article on proper time located here.

We might also discuss the "proper length" between events rather than proper time. That formulation leads to those equations with the negative signs.

http://en.wikipedia.org/wiki/Proper_length

[quote]In special relativity, the proper length between two spacelike-separated events is the distance between the two events, as measured in an inertial frame of reference in which the events are simultaneous. So if the two events occur at opposite ends of an object, the proper length of the object is the length of the object as measured by an observer which is at rest relative to the object.[\quote]

Did you even bother to look at the example calculations in those wikipedia articles I recommended?

Yes, but when distance is measured in light-years and time is measured in years then c squared = 1.

Don't be silly. There are no proper length examples. However the discussion in the article answers the question you asked me. If you believe I've provided you with an improper equation, I'll be happy to acknowledge and correct that.

With c² = 1, the equation appears to be exactly the same as what I posted. Thanks we can do away with that accusation.

By discussion I definitely don't mean what's posted on the talk page. Instead I mean the discussion that talks about the meaning of proper length and proper time.

Imagine an object moving relative to the earth. The observed length of the object varies with the velocity relative to earth, but the proper length is independent of the relative velocity. The proper length then is a property of the object, while in general, the observed length is not.

I yield. This is not fun anymore, and I usually enjoy talking about relativity.

Edited by NoNukes, : No reason given.