Minkowski 4-space? Yes or No.

[stevmg]

In Einstein's "Relativity," this equation appears in Appendix I (Lorentz Transformations):

x₁'² + y₁'² + z₁'² - c²t₁'² = x₁² + y₁² + z₁² - c²t²

This essentially says that for a given event at a given point and time in space, looking at this event from any frame of reference will give results that satisfy this equation for that event. In other words, x₁² + y₁² + z₁² - c²t² is a constant.

I assume that t is not a parameter but merely a coordinate, just as x, y and z are.

I am used to, from analytic geometery and linear algebra, that in the case of a parameter, once a value is assigned, will map to, simutaneously, all the coordinates and, since there is only one value in each coordinate that a given parametric value can map to, then we are essentially describing a line (straight or curvilinear) in space. The above equation, to my thinking, does not represent a parametric equation as the t can map to several x, y and z'sz once a value for it is assigned.

Do we jump off here to "world lines?" or is this completely different?

 

[tiny-tim]

I am used to, from analytic geometery and linear algebra, that in the case of a parameter, once a value is assigned, will map to, simutaneously, all the coordinates and, since there is only one value in each coordinate that a given parametric value can map to, then we are essentially describing a line (straight or curvilinear) in space. The above equation, to my thinking, does not represent a parametric equation as the t can map to several x, y and z'sz once a value for it is assigned.

Do we jump off here to "world lines?" or is this completely different?

Hi stevmg!

That equation is not a parametric equation of a line or any other curve, it is a general equation mapping the whole of space-time onto itself.

It is a rotation in 4-space, just like any ordinary rotation in 3-space.

World-lines are completely different, they are curves in space-time (with "gradient" either always less than c, or always equal to c).

 

[Jonathan Scott]

That equation is normally used to describe the fact that the magnitude of the space-time separation between two events in space-time does not depend on the frame of reference. The variables x,y,z,t are the displacements and time interval between the two events in one frame and the primed variables are the displacements and time interval as measured in any other frame.

 

[bcrowell]

Stevemg, since you're trying to visualize this in geometric terms, a good case to visualize is the one where these two expressions are both equal to zero. In that case you have the equation of a four-dimensional cone, which is the light cone: http://en.wikipedia.org/wiki/Light_cone

 

[Naty1]

World lines originate from the Einstein field equations...
try here
http://en.wikipedia.org/wiki/World_lines

 

[Jonathan Scott]

For that equation to hold for general events, the two sets of coordinates must both be measured relative to a common origin event. In that case, the space-time location of any event satisfies that equation; it says that the space-time distance between the event and the origin is independent of (special relativity) inertial reference frame.

 

[Dale]

The previous responses are correct. I will only add that it is certainly possible to write a parametric equation of a world line, e.g.:

$$\left(c t(\lambda), x(\lambda), y(\lambda), z(\lambda)\right)$$

Where $\lambda$ is any arbitrary parameter, not necessarily equal to either coordinate or proper time.

 

[stevmg]

Thank you for coming in, DaleSpam...

So the equation I presented is NOT a parametric equation after all (I thought not.) I like response #6 Jonathan Scott description of it. In other words, my description of that equation is correct - it describes an event in x, y, z and t and any and all combinations (4-tuples) of x, y, z and t which add up to the same value are merely ways describing that same event from another frame of reference. Sort of like x² + y² + z² = 25. Any and all values of x, y and z (3-tuples) that satisfy that equation are points on a sphere of radius 5 and the center at the origin (0, 0, 0.)

Now, DaleSpam has defined a "world line" parameteric system using $$\lambda$$ as the arbitrary parameter (not necessarily a coordinate or point in time) that will define the world line or curve in space-time.

Question, as I am a little fuzzy on parametric equations: Is the parametric depiction a one-to-one function in which $$\lambda$$ is the domain for the coordinates (ct($$\lambda$$), x($$\lambda$$, y($$\lambda$$), z($$\lambda$$)) where each coordinate is the range? Again, if so, then what is defined would be some curvilinear line in, say, 4 dimensions (given those coordinates above). In other words, for each $$\lambda$$, there is only ONE value for each coordinate (of course, two different $$\lambda$$'s can map to the same, say x, but not the other way around.)

I am aware of the "light cone" and I visualize it (using, say, just x and ct) as two lines crossing at the origin (like an "X") in which all points in the real world lie above the right ("/") land the left ("\") lines. It is not possible to to lie to the right or left of the "X" (outside the cone.)

If, by some miracle, one could get "outside the X" (outside the cone), then one could observe events happening in reverse order. In the real world, if event A precedes event B, this is true in all frames of reference and even in all the $$\tau$$ frames of reference(the frame of reference does not move, only time changes.) Getting "outside the cone" allows one to look at the "real world" and see event B precede event A.

Am I on the right track? Are there, theoretically, parametric systems of equations to describe all world lines which are "within the cone?"

 

[Dale]

Question, as I am a little fuzzy on parametric equations: Is the parametric depiction a one-to-one function in which $$\lambda$$ is the domain for the coordinates (ct($$\lambda$$), x($$\lambda$$, y($$\lambda$$), z($$\lambda$$)) where each coordinate is the range? Again, if so, then what is defined would be some curvilinear line in, say, 4 dimensions (given those coordinates above). In other words, for each $$\lambda$$, there is only ONE value for each coordinate (of course, two different $$\lambda$$'s can map to the same, say x, but not the other way around.)

That is all correct.

I am aware of the "light cone" and I visualize it (using, say, just x and ct) as two lines crossing at the origin (like an "X") in which all points in the real world lie above the right ("/") land the left ("\") lines. It is not possible to to lie to the right or left of the "X" (outside the cone.)

If, by some miracle, one could get "outside the X" (outside the cone), then one could observe events happening in reverse order. In the real world, if event A precedes event B, this is true in all frames of reference and even in all the $$\tau$$ frames of reference(the frame of reference does not move, only time changes.) Getting "outside the cone" allows one to look at the "real world" and see event B precede event A.

This is also esentialy correct. One other way to look at the light cone is in terms of the spacetime interval equation you gave in your OP: s² = x² + y² + z² - c²t². Events inside the cone will have a negative s² (aka timelike), events outside the cone will have a positive s² (aka spacelike), and events on the cone will have s²=0 (aka lightlike or null).

Am I on the right track? Are there, theoretically, parametric systems of equations to describe all world lines which are "within the cone?"

Yes, such a parametric equation is called timelike. That means that $(ds/d\lambda)^2<0$

 

[stevmg]

Thanks...

Again, I am approaching my brain limit.