Are motion through space and time inversely proportional?

[Fanghur]

I've always thought that speed through space was inversely proportional to speed through time, and vice versa. In other words the faster you move through through space, the slower you move through time.

 

Is this analogy really true? I mean, it's true as you approach the speed of light, but is it also true for all speeds?

[timo]

Whatever inverse proportionality is supposed to mean I imagine problems for [math]\vec v = \vec 0 [/math].

 

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The relation between space-distance and time-distance travelled during an eigentime interval is like [math]v_t^2 = \vec v^2 / c^2 + 1[/math] for massive particles.

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Edit note: I incorrectly mixed [math]\vec v = \frac{d\vec x}{dt}[/math] and [math]v_t = \frac{dt}{d\tau}[/math] above (not the different variables against which the differential is taken).

[swansont]

The four-vector stays the same length. The faster you travel, the more your distance is length contracted, and the more your clock is slowed, relative to another observer. The factor for both changes is gamma, so in that sense they are inverses.

[Fanghur]

Let me rephrase the original question. Let's say you are moving at 100% the speed of light, if this was the case your clock would be not moving at all, in other words you would be moving through time 0%. Now let's say you are travelling at 50% the speed of light, then your clock would be running 50% slower relative to someone at rest. My original question was whether this relationship between space and time was linear, or is it more complicated that just being inversely proportional?

[Hypercube]

That's how I picture the time dilation effect too, but I don't think that they are linearly inversely proportional, that's just an easy way to think about it.

[Graviphoton]

Am i undestanding this correctly... are you stating movement of a body is ''invariant'' with space and time - in other words, must a body move through both space at the same time as time? The use of the words inversely proportional maybe be very decieving...

 

But here is the answer.

 

Remove the notion of space and time, and make it one system, spacetime. Now you can only move through the one continuum. With different speeds, an object can travel through different mathematical vectors of spacetime... for instance, a particle travelling below c, is found to move through real space, or imaginary time. A photon is squeezed out of existence, because it moves at c, therefore it experiences no passing of time, or ofcourse, movement through spacetime. Its birth and death are simultaneous. And then there are tachyons, which contain an infinite amount of energy at their lowest speeds c and using as little energy as possible over c... These particles move through imaginary space, or real time.

 

I hope this helps.

[timo]

If an object moves with a velocity v<c in some coordinate system then the ratio between time passage in this coordinate system and the time passage for the particle equals the Gamov factor [math]\gamma = \frac{1}{\sqrt{1-v^2/c^2}}[/math] (which is the same gamma as the one mentioned by Swansont). Gamma is neither proportional nor anti-proportional to v.