The Immovable Universe: Exploring Objects at Rest

[chaszz]

Is there any physical object in the universe that is not in motion?

 

[jtbell]

Every object in the universe is at rest with respect to itself.

 

[Mentz114]

Is there any physical object in the universe that is not in motion?

In motion with respect to what ? We can only usefully define relative motion.

 

[Chestermiller]

Is there any physical object in the universe that is not in motion?

Some people (myself included) like to think of time as an actual spatial direction (and coordinate) in 4D Minkowski space. This often provides a useful conceptual model for envisioning what's going on. However, in Minkowski space, the "time direction" does not, in all respects, share exactly the same characteristics with the three spatial directions. If 4D spacetime were Euclidean, rather than Minkowskian, then time could truly be regarded as a spatial direction on par with the other three spatial directions. But it is not, and so, there are some differences. What are the differences?

In both Minkowski space and Euclidean space, an absolute 4D position vector s (event vector) relative to an arbitrary origin can be represented by:

s = cti_t+xi_x+yi_y+zi_z

where the boldface i's in this equation represent basis vectors in the coordinate directions. In 4D Euclidean space, the dot product of each basis vector with itself is equal to +1, and the dot product of each basis vector with the other basis vectors is equal to zero. In Minkowski space, the main difference is that dot product of the time basis vector with itself is equal to -1.

For two inertial frames of reference in relative motion, S and S', the dot products of the spatial basis vectors for the S frame of reference with the spatial basis vectors for the S' frame of reference are all less than unity (in magnitude), and equal to the direction cosines. However, the dot products of the time basis vectors are negative and greater in magnitude than unity (the relativity factor). More importantly, the time basis vector for the S' frame of reference is a linear combination both of the time basis vector and the spatial basis vectors for the S frame of reference, and the spatial basis vectors for the S' frame of reference are a linear combination both of the time basis vector and the spatial basis vectors for the S frame of reference. Thus, the time basis vector for the S frame of reference has components in the spatial directions of the S' frame of reference. In this sense, the time basis vector and direction possesses a kind of spatial quality, even in Minkowski space.

In line with the above discussion, if we consider any arbitrary object in Minkowski space, its "absolute" position vector s relative to an arbitrary origin in spacetime, can be expressed in terms of the basis vectors for its rest frame of reference by:

s = cti_t

where its position within its own rest frame has been taken to be x=y=z=0. According to this equation, the object is traveling through absolute spacetime at the speed of light. This same analysis can be applied to all objects in spacetime. The only difference is the directions of their time basis vectors. In my opinion, thinking of the time direction in this way makes SR geometry and concepts easier to visualize. And, it will not lead to the wrong answers in solving actual physical problems.

 

[Ich]

For two inertial frames of reference in relative motion, S and S', the dot products of the spatial basis vectors for the S frame of reference with the spatial basis vectors for the S' frame of reference are all less than unity (in magnitude), and equal to the direction cosines.

For a Lorentz boost along one axis x, the dot product of x and x' ist greater than unity, while for y and z the dot product is 1. The behaviour you described applies for spatial rotations, not boosts.

In my opinion, thinking of the time direction in this way makes SR geometry and concepts easier to visualize. And, it will not lead to the wrong answers in solving actual physical problems.

I agree in principle, but I'm not sure if your answer will be helpful for the OP.

 

[Chestermiller]

For a Lorentz boost along one axis x, the dot product of x and x' ist greater than unity, while for y and z the dot product is 1. The behaviour you described applies for spatial rotations, not boosts.

Yes. Thanks for the helpful correction. Of course, I knew that; I don't know what I was thinking. Sorry for any confusion I might have caused.

Chet