Time a fourth dimension? is a boost a rotation?

[jrrship]

is a boost a rotation of a four vector?

if the boost rotates an object more into the time dimension, ywo things happen:

it's velocity increases.
it's length contract in the direction of its motion.

does this make sense?

thanks!

 

[dextercioby]

In the usual parametrization, one can see that a Lorentzian boost in the "x" direction can be seen as a rotation in the plane (t,x) by an complex imaginary angle.

Daniel.

 

[jrrship]

So does that mean that a boost in the x direction means that the opbject is rotated more into the t direction--does it mean that it has a greater tc omponent, and a smaller x component?

Where are the best resources that talk about boosts?

Thanks!

 

[Mortimer]

You may want to have a look at https://www.physicsforums.com/showthread.php?t=103977" on Euclidean Special Relativity in the Independent Research forum. In Euclidean relativity (which is not generally accepted by the way and is inconsistent in some details with standard relativity) boosts are indeed rotations in 4D space-time.

Rob

 

[jrrship]

thanks rob--i'd prefer to stick with topics that are consistent with relativity.

So does that mean that a boost in the x direction means that the object is rotated more into the t direction--does it mean that it has a greater t component, and a smaller x component? is that why it appears shorter?

Where are the best resources that talk about boosts?

Thanks!

 

[HallsofIvy]

Could someone please define the word "boost" as it is used here. I don't recognize it.

 

[jrrship]

I find it puzzling that a PF mentor with over 10,000 posts has never heard of a boost. Have you ever taken a course on SR?

The way I see it, a "boost" is a rotation in four dimensional space-time.

 

[tim_lou]

i don't think it is a real rotation, it's just that if you treat the rapidity like an angle, the Lorentz transformation in terms of rapidity is similar to the equation of rotating except sin becomes sinh and cos becomes cosh (in terms of hyperbolic function).

well, I guess the lorentz transformation does preserve dot product and the determinate does indeed come out to 1... I guess you can call it a rotation in terms of linear algebra.

 

[Chris Hillman]

Boosts versus rotations

is a boost a rotation of a four vector?

No, but boosts and rotations are linear transformations acting on Minkowski spacetime, in fact Lorentz transformations.

Boosts are analogous to rotations but they are not rotations (at least, not in real Lorentzian manifolds).

You can try the textbook by Tristam Needham, Visual Complex Analysis, which happens to have a very nice discussion of the Lorentz group and its Moebius action on the celestial sphere.

 

[jrrship]

My question is this.

A boost, acting on an object, takes a component out of the spatial dimensions and gives it a greater presence in the time dimension.

Hence it appears shorter as it moves.

Is this not true?

 

[Chris Hillman]

Well, I hesitate to say "yes" or "no" because I don't know what you mean by this verbal description. Some comments:

1. "takes a component out of the spatial dimensions": you probably are thinking of a spacelike hyperplane element and using a boost to boost a vector lying in some an element so that it points out of the hyperplane,

2. "gives it a greater presence in the time dimension": you probably mean "increases its time component".

3. "it appears shorter": "appearance" is tricky in this context. If you mean "visual appearance", this would not be obtained by a simple Lorentz transformation.

 

[jrrship]

Yes--correct me if I am wrong.

A boost that reduces an object's spatial component in the x direction increases its time component.

This is true, right?

Thanks!

 

[Chris Hillman]

No, I think you are struggling to describe in words (a mathematical description would be a better idea!) how a rotation affects components of a vector. Boosts are analogous but behave a bit differently. See Taylor and Wheeler, Spacetime Physics, First Edition only (because you need the notion of rapidity which was dropped from the second edition).

 

[jrrship]

You say,

"Boosts are analogous but behave a bit differently."

How?

 

[Mortimer]

A boost that reduces an object's spatial component in the x direction increases its time component.

Perhaps you should have a look at Euclidean SR after all (e.g. http://www.euclideanrelativity.com/simplified/index.htm#four" ). I'm sure you'll find it interesting, despite it being non-mainstream. Sorry for insisting.

Rob

 

[Chris Hillman]

Mortimer, I fear that jrrship will only be confused if he doesn't first master the basic ideas of Lorentzian geometry. Among other things that involves mastering hyperbolic versus parabolic versus elliptical trig (the last is ordinary high school trig, the one before that is used in "Newtonian spacetime" a la Cartan, the first is of course the basis for the kinematics of str). So to some extent this is a matter of emphasis and interpretation.

jjrship, you asked how boosts differ from rotations. Well, as I said, both boosts and rotations (acting on Minkowski spacetime) are special cases of Lorentz transformations. (There are many "loxodromic" LTs which are neither a boost or a rotation, but can be built up from boosts and rotations by composition of transformations.) All Lorentz transformations have the property that they always transform a spacelike, null, or past/future pointing timelike vector to a spacelike, null, or past/future pointing timelike vector respectively. Now, a rotation can "turn a spacelike vector all the way around in space", so that through any event in any small neighborhood there exist circles, closed spacelike curves. But a boost cannot "turn a future pointing timelike vector around to become past pointing timelike vector"; closed timelike curves do not exist in Minkowski spacetime. This is perhaps the most fundamental geometric difference.

 

[robphy]

This may be helpful: google spacetime trigonometry (shameless plug?)