# I Rotation or acceleration defined without relation to something else

#### cianfa72

Summary
understanding how rotation or acceleration of an object can be defined without any relationship between the object itself and something else
Starting from this post, we are able to define the concept of (proper) acceleration or rotation without any reference to something else
It's defined in terms of accelerometers and gyroscopes. Basically, you set up three gyroscopes whose axes point in three mutually orthogonal spacelike directions. Then you set up accelerometers to measure acceleration in each of those three directions. Then you carry along this apparatus next to the object, so that you can watch the readings of the accelerometers and the relationship between the spatial orientation of the object and the axes of the gyroscopes. Nonzero accelerometer readings means "acceleration"; change in the orientation of the object relative to the gyroscopes means "rotation".
About this definition which is the physical meaning of gyroscopes axes pointing in three mutually orthogonal spacelike directions ?
In other words, from a physical point of view, how can we select those three mutually orthogonal spacelike directions ?

Thanks

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#### jbriggs444

Homework Helper
In other words, from a physical point of view, how can we select those three mutually orthogonal spacelike directions ?
The Egyptians did it with knotted rope and 3-4-5 triangles if I remember correctly.

Edit: Seems the Babylonians are known to have calculated pythagorean triples circa 2000 BC.

Personally, I'd go to the hardware store and buy a framing square.

#### Dale

Mentor
Personally, I'd go to the hardware store and buy a framing square.
Or a speed square. It is probably my third favorite tool after my chainsaw and impact driver. Anyway, @cianfa72 should definitely pick up those at the hardware store too. The chainsaw may not be useful for this project but it is just too much fun to pass up!

#### cianfa72

Or a speed square.
Sure, but actually the point was: I'm aware of worldlines of any real object are timelike. Why the three directions singled out that way are taken to be spacelike ?

#### Ibix

Why the three directions singled out that way are taken to be spacelike ?
Because they must be. Vectors orthogonal to the timelike direction can't have aa component in the timelike direction.

It's a bit like asking why vectors perpendicular to the vertical are horizontal.

#### Dale

Mentor
Sure, but actually the point was: I'm aware of worldlines of any real object is timelike. Why the three directions singled out that way are taken to be spacelike ?
The directions of the axes are not along the worldline of the device.

#### cianfa72

The directions of the axes are not along the worldline of the device.
As per my understanding, to single out a direction in spacetime we have to take in account two "near" events defining the "four-vector" joining them. For the device (gyroscope) worldline's events the four-vectors joining them are timelike but what about for the direction of one of that three axes ? Which kind of "near" events we have to consider for it ?

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#### pervect

Staff Emeritus
Summary: understanding how rotation or acceleration of an object can be defined without any relationship between the object itself and something else

Starting from this post, we are able to define the concept of (proper) acceleration or rotation without any reference to something else

About this definition which is the physical meaning of gyroscopes axes pointing in three mutually orthogonal spacelike directions ?
In other words, from a physical point of view, how can we select those three mutually orthogonal spacelike directions ?

Thanks
The meaning of spacelike separations is that two points that occur "at the same time", two points that are synchronized in time, are separated by space with no diffrence in time.

Another way of saying this is that "space" is the set of all events in space-time that happen "at the same time".

"At the same time" is defined by the Einstein clock synchronization convention. Violating this convention makes physics non-isotorpic. For a real world example, consider what might happen if one took the idea that it took an airplane 3 hours to fly from the east coast of the USA to the west coast, but 9 hours to fly back, seriously. One would have to explain why the airplanes flew faster in one directions - this would be a failure of isotropy.

What would the "physical" reason be for identical airplanes to fly faster in one direction than the other?

Generaly, isotropy is assumed, and this implies that one uses the Einstein convention for synchronizing clocks. And once one has a clock synchronization convention, one can define "space" as the set of events that happen simultaneously.

#### Dale

Mentor
For the device (gyroscope) worldline's events the four-vectors joining them are timelike
The device is an extended body, ie a worldtube rather than a worldline. It has spacelike directions, such as the axes formed by constructing it with the speed square.

#### PeterDonis

Mentor
to single out a direction in spacetime we have to take in account two "near" events defining the "four-vector" joining them
No, we don't. 4-vectors are defined at a single point--more precisely, in the tangent space to spacetime at a single point. Each point has a different tangent space. The concept of a vector as an "arrow" connecting nearby points is often used to introduce the topic, but it is not precisely correct, and it's worth unlearning it and learning the correct concept.

#### sweet springs

Summary: understanding how rotation or acceleration of an object can be defined without any relationship between the object itself and something else

we are able to define the concept of (proper) acceleration or rotation without any reference to something else
Literally rotation or acceleration belongs to motion. Motion belongs to relationship with other body or bodies in space and time. So the question posed includes contradiction.

#### Dale

Mentor
Literally rotation or acceleration belong to motion. Motion belongs to relationship with other body or bodies in space and time. So the question posed includes contradiction.
No, this is not correct. Velocity is relative. That does not imply that all motion is relative. Rotation and acceleration are not.

#### cianfa72

No, we don't. 4-vectors are defined at a single point--more precisely, in the tangent space to spacetime at a single point. Each point has a different tangent space. The concept of a vector as an "arrow" connecting nearby points is often used to introduce the topic, but it is not precisely correct, and it's worth unlearning it and learning the correct concept.
Sure that's right. Nevertheless I'm a bit confused about the physical content to "attach" to a spacelike vector belonging to an event's tangent space. Regarding timelike/null vectors there is no doubt: they represent the worldlines of all massive particles (timelike) or light rays (lightlike - null separated) passing through the given event.

What about the physical content for spacelike vectors at a given event ?

I'm aware of in the context of Minkowski flat spacetime a spacelike vector at a given event is the spacetime direction from the given event to reach a nearby event for which we can find an inertial frame of reference at he given event in which the Einstein convention synchronized clocks reading for the two event are the same (here the concept make sense due to the affine structure of flat spacetime). Can we apply the same argument in case of general curved spacetime ?

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#### Ibix

What about the physical content for spacelike vectors at a given event ?
Three orthogonal spacelike vectors at an event together define a spacelike "plane" which contains all possible local notions of "space at the same time" that include that event. This is always true. The special thing about Minkowski spacetime is that you can trivially stitch these local definitions of "space" into a global definition of "space". In curved spacetime there's usually more than one way to do this stitching, and it's often impossible to extend a given methodology to cover the entire spacetime.

#### sweet springs

Velocity is relative. That does not imply that all motion is relative.
I do not understand your point. Aren't all motion relative as well as velocity is relative? Here I do not mean TOR things, e.g. IFR or fictitious force, but mind very basic concept of motion or velocity of the thing that could stand in relationship with something other than itself, I may say it "against what".

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#### Dale

Mentor
I do not understand the point. Isn't motion relative as well as velocity is relative?
The point is that velocity is relative but acceleration is not relative. The broad term “motion” includes both velocity and acceleration so the broad statement “motion is relative” is not correct.

Inside a box, with no means of interacting with the outside, there is no measurement you can make which will detect your velocity. There are measurements you can make which will detect your acceleration and rotation and any higher order motion.

This aspect of the OP’s question is not a contradiction as you incorrectly asserted above.

#### phinds

Gold Member
I do not understand your point. Aren't all motion relative as well as velocity is relative? Here I do not mean TOR things, e.g. IFR or fictitious force, but mind very basic concept of motion or velocity of the thing that could stand in relationship with something other than itself, I may say it "against what".
Acceleration is not relative. If you are in an accelerating spaceship you can stand on a scale and see that there is acceleration and this is utterly unrelated to anything outside the spaceship. Similarly, rotation is inherently acceration and is not relative.

EDIT: I see Dale beat me to it.

#### cianfa72

Three orthogonal spacelike vectors at an event together define a spacelike "plane" which contains all possible local notions of "space at the same time" that include that event. This is always true.
Those local notions of "space at the same time" are basically due to the ability to single out a set of local inertial frames at a given event each of them endowed with its own set of clocks at rest synchronized according Einstein convention ?

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#### metastable

It’s interesting the 3 axis’ we use to reference spatial dimension consist of straight lines while in my understanding no real particle follows such a path. Could I describe a curved path without an x-y axis by specifying the radius of a circle whose circumference matches the local curvature?

#### PeterDonis

Mentor
Regarding timelike/null vectors there is no doubt: they represent the worldlines of all massive particles (timelike) or light rays (lightlike - null separated) passing through the given event.
More precisely, they represent tangent vectors to such worldlines--in other words, they represent all of the possible directions in spacetime that such worldlines can go.

What about the physical content for spacelike vectors at a given event ?
They represent all of the possible directions in spacetime that spacelike curves can go. @Ibix explained what kinds of things, physically, spacelike curves connect.

in the context of Minkowski flat spacetime a spacelike vector at a given event is the spacetime direction from the given event to reach a nearby event
No. The same thing I said before about timelike vectors applies here as well: vectors are not "arrows" between one event and a nearby event.

#### pervect

Staff Emeritus
Those local notions of "space at the same time" are basically due to the ability to single out a set of local inertial frames at a given event each of them endowed with its own set of clocks at rest synchronized according Einstein convention ?
There might be some minor quibbles, but basically yes. For some context, I'm imaging that we have a general, not necessarily flat, space-time, that we represent by a mathematical structure called a manifold. Then the local inertial frame near some point in the space-time would be called the tangent space of the manifold, when we represent the space-time with the mathematical structure we call a manifold. Every point in the general space-time would have it's own local inertial frame, just as every point on the manifold has its own tangent space.

#### Nugatory

Mentor
Every point in the general space-time would have it's own local inertial frame.
Actually that would be multiple inertial frames, is it not? Consider the worldlines of two different objects that intersect at that point... there’s one tangent space at that point but the objects are at rest in different inertial frames.

#### pervect

Staff Emeritus
Actually that would be multiple inertial frames, is it not? Consider the worldlines of two different objects that intersect at that point... there’s one tangent space at that point but the objects are at rest in different inertial frames.
A good point - if we have multiple observers, moving at different velocities, but happen to occupy the same point at the same time, they have different local inertial frames, but they share the same tangent space. So the inertial frame has a bit more structure than the tangent space.

Technically, we create an inertial frame of reference from a tangent space by choosing a set of basis vectors. I was hoping to avoid this level of complexity, but I suppose it was wrong-headed to try and simplify it too much.

"Rotation or acceleration defined without relation to something else"

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