Is motion through space or spacetime?

[Dale]

Sounds reasonable to me except that I would say "one could" rather than "one would". The point is that either interpretation is a matter of choice and the physicist should feel at liberty to pick either, both, or neither as occasion suits. The universe doesn't care either way.

 

[Frank Castle]

Sounds reasonable to me except that I would say "one could" rather than "one would". The point is that either interpretation is a matter of choice and the physicist should feel at liberty to pick either, both, or neither as occasion suits. The universe doesn't care either way.

Good point.

 

[vanhees71]

I would not agree with this particular part of the quoted text. The length of the tangent vector for massive particles does have an obvious physical interpretation: the rest mass of the particle. Knowing that a particle has a particular worldline in spacetime does not tell us all there is to know about the particle; it has other properties, of which rest mass is one, and the length of the tangent vector models that property.

I also like to use the "normalized" four-velocity vector
$$u^{\mu}=\frac{\mathrm{d} x^{\mu}}{\mathrm{d} s}=\frac{1}{c} \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau}=\frac{1}{mc} p^{\mu},$$
just because it's convenient, particularly in continuum mechanics it's nice to have the normalized flow-velocity vector to project between the temporal and spatial components of four vectors wrt. the local restframe of the heat bath, using the covariant projectors
$$P_{\parallel}^{\mu \nu}=u^{\mu} u^{\nu}, \quad P_{\perp}^{\mu \nu}=\eta^{\mu \nu}-u^{\mu} u^{\nu}.$$

For the motion of a point particle one should be aware that the covariant equation of motion only superficially has four independent components
$$\frac{\mathrm{d} p^{\mu}}{\mathrm{d} \tau}=K^{\mu},$$
because you have (for classical particles!) the on-shell contraint
$$p_{\mu} p^{\mu}=m^2 c^2 \; \Rightarrow \; p_{\mu} \frac{\mathrm{d} p^{\mu}}{\mathrm{d} \tau}=0 \; \Rightarrow \; p_{\mu} K^{\mu}=0.$$
Thus only three of the four equations are independent. You can solve for the three spatial components in the respective frame of reference ("calculational frame"), and the time-component follows. It's in some sense a generalized energy-work theorem:
$$u_{\mu} K^{\mu}=0 \; \Rightarrow \; u^0 K^0=\vec{u} \cdot \vec{K} \; \Rightarrow \; K^0=\frac{\vec{u}}{u^0} \vec{K} = \frac{\vec{v}}{c} \cdot \vec{K}.$$
So you have
$$\frac{\mathrm{d} p^0}{\mathrm{d} \tau}=\frac{1}{c} \frac{\mathrm{d} \mathcal{E}}{\mathrm{d} \tau}=K^0=\frac{\vec{v}}{c} \cdot \vec{K}.$$
Now you can rewrite this in terms of the coordinate time, using $\mathrm{d} t/\mathrm{d} \tau=u^0=\gamma$.
$$\frac{\mathrm{d} \mathcal{E}}{\mathrm{d} t}=\vec{v} \cdot \vec{F}, \quad \vec{F}=\frac{\vec{K}}{u^0}.$$

 

[PAllen]

So globally one would interpret spacetime as being static, in the sense that time does not flow within it, time is itself a coordinate labelling events (along with 3 spatial coordinates) of spacetime, hence any flow of time would require re-introducing time when it has already been "used up" to construct the spacetime in the first place. In this sense, the worldlines of objects exist throughout spacetime and are not themselves moving through spacetime - everything is static. However, locally, objects with timelike worldlines can be ascribed a proper-time which parametrises their worldline. Since proper-time is fundamentally distinct from coordinate time - it is frame independent - we can utilise it as a "meta-time" and in this sense interpret a timelike object as locally propagating through spacetime from point to point along its worldline and constant speed $c$?!

Yes, except there is not much meaning to speed c. For example, personally, I normalize 4-velocities to 1 rather than c, even when I am not setting c=1. There is no physics whatsoever in varying choice of affine parameter (and you can get any positive 'speed' you want by suitable choice of affine parameter; you can choose 42 if you are galactic hitch-hiker).

 

[vanhees71]

Sure $c$ is only a conversion factor from inconvenient SI units to physical units (SCNR).

 

[Jeronimus]

If the universe is a static 4 dimensional construct and there is no motion to it, then you would have to explain why we experience change still.

We certainly do not experience the whole universe at the same time.
The universe might be static and not moving, but our experience of it is moving along the worldlines the instance of our bodies reside in, in what we call the present.

 

[Dale]

If the universe is a static 4 dimensional construct and there is no motion to it, then you would have to explain why we experience change still.

That is easy to explain. At each event on your worldline your experience is a function of the past light cone of that event. Since the past light cone changes along the worldline the experience also changes along the worldline.

We certainly do not experience the whole universe at the same time.

Of course not, most of the universe is outside the past light cone of any given moment, and even within our past light cone the closer events are more important in determining our experience.

The universe might be static and not moving, but our experience of it is moving along the worldlines the instance of our bodies reside in, in what we call the present.

At every event you have the experience of being present. So what we call "the present" is not a unique physically identifiable moment, but rather a psychological impression that applies equally to any moment.

 

[Frank Castle]

Of course not, most of the universe is outside the past light cone of any given moment, and even within our past light cone the closer events are more important in determining our experience.

This would be why one can locally interpret a timelike object as propagating along its worldine and hence moving in spacetime, right? (if we could observe the universe as a whole, neglecting gravity, then one could observe the entire worldline of the timelike object and hence the object would be at all points along its worldline).

 

[Mister T]

We certainly do not experience the whole universe at the same time.

We don't experience the whole universe at ANY time. Every experience I ever have occurs at one specific place AND at one specific time. Experience is a very limited and personal thing.

 

[Kevin McHugh]

There are some very enlightening responses in this thread. I'd like to share how this dumbazz imagines a world line in space time. I think of calculus, and the moving triad of tnagent, normal, and binormal vectors moving along a parametized curve. In this case of SR, the curve is a function of proper time, and the the triad is 3D Euclidian space. You can think of a particle at rest within that space, moving along at proper time, or the particle can have momentum within that space, as well as moving along the timeline (i.e. world line). Just my $0.02

 

[Ebeb]

As a personal preference, while some may think of a world-line (-surface, -tube) as a mere path or trajectory of an object through space-time I like to think of it as being the object itself, a 4D object. What we perceive as the "object" is then just a slice of the actual 4D object intersected with an arbitrary 3D "now" surface. The 4D object itself is fixed and unchanging in this view.

Hi Vitro. You look at it the way Einstein did. Here are a few Einstein quotes you might find most interesting:

<< Since there exists in this four dimensional structure [space-time] no longer any sections which represent "now" objectively, the concepts of happening and becoming are indeed not completely suspended, but yet complicated. It appears therefore more natural to think of physical reality as a four dimensional existence, instead of, as hitherto, the evolution of a three dimensional existence. >> (Albert Einstein, "Relativity", 1952).

<< From a "happening" in three-dimensional space, physics becomes, as it were, an "existence" in the four-dimensional "world". >> (Albert Einstein. "Relativity: The Special and the General Theory." 1916. Appendix II Minkowski's Four-Dimensional Space ("World") (supplementary to section 17 - last section of part 1 - Minkowski's Four-Dimensional Space).