Can someone clarify and shed some light about these two statements

[jrist29]

If every dynamic variable in a physics problem is a measurement derived from a particular coordinate system how do you explain the contradiction both statements 1 and 2 imply if they are both simultaneously true:

1. According to relativity (relative velocity etc.) every measurement of position, distance, displacement, velocity and acceleration is dependent on the particular coordinate system used to make the measurements and by that logic every observer should get a different measurement based on the particular coordinate system used (i.e. placement of origin and orientation of axes)
2. According to transformation laws and tensor analysis vectors and scalars are covariant or form invariant and therefore measurements do not depend on the particular coordinate system used and by that logic every observer should get the same measurement regardless of their particular frame of reference
Unless I have grossly misunderstood physics and how we acquire information about the world (which is more than likely the case) there seems to be a major contradiction here. Correct me if I am wrong please because I do not see a way out of this conundrum!

 

[Chestermiller]

What 1 refers to are the components of vectors and tensors as reckoned from various coordinate systems. As you know, the vectors themselves can be expressed as the sum of their components times the unit vectors in the coordinate directions. If you change frame of reference (coordinate system), you change the unit vectors, and, that causes the vector components to change. But the sum of the components times the unit vectors remains the same.

Item 2 refers to the vectors themselves (not their components). Of course, the vectors themselves do not change when you change frame of reference.

 

[jrist29]

Thanks Chestmiller! I have taken quite a few physics courses and the all the information starts to blur after awhile if I don't keep the concepts straight lol! I guess I need to practice more problems! But you hit the nail on the head by clarifying the distinctions between the actual vector and the vector representation in a particular coordinate system. I guess as I was learning physics that concept was not made clear enough to me or I didnt give it the attention it deserved.

Thanks a million
jrist29

 

[HallsofIvy]

As a very very simple example, If I set up a Cartesian coordinate system in a plane, the vector from the origin, (0, 0), to the point (1, 1) is, of course, [itex]\vec{i}+ \vec{j}[/tex]. If I rotate that coordinate system so that my new x-axis was pointing along the line y= x (in the original coordinate system) then that same vector would be represented by $\sqrt{2}\vec{i'}$ where, now, $\vec{i'}$ is the unit vector along the new x-axis. The components have changed, but it is still the same vector. (And, in either coordinate system, it length is "invariant".)

 

[sardar]

I can understand how these two statements may seem contradictory at first glance. However, upon further examination, they are actually both true and do not contradict each other.

First, let's clarify what each statement is saying. Statement 1 is referring to the concept of relativity, which states that all physical laws are the same for all observers in uniform motion. This means that different observers may measure different values for position, distance, displacement, velocity, and acceleration, but these values are all equally valid as long as they are measured from their own particular frame of reference. This is because the measurements are dependent on the particular coordinate system used, as stated in the statement.

On the other hand, statement 2 is referring to the concept of transformation laws and tensor analysis. These mathematical tools allow us to transform measurements from one coordinate system to another, without changing the underlying physical laws. This means that while different observers may use different coordinate systems, the measurements they obtain can be transformed into each other and will still be consistent and valid. This is because vectors and scalars are covariant, meaning they maintain their properties under coordinate transformations.

So, how do we reconcile these two seemingly contradictory statements? The key is to understand that they are referring to different aspects of the same phenomenon. Relativity deals with how measurements are affected by the observer's frame of reference, while transformation laws deal with how measurements can be transformed between different coordinate systems.

In other words, statement 1 is describing the subjective nature of measurements, while statement 2 is describing the objective nature of the underlying physical laws. Both statements are true and do not contradict each other. It is important to note that while the measurements may be different for different observers, the physical laws governing those measurements remain the same.

I hope this explanation has clarified the issue and helped to resolve any confusion. As scientists, it is important to understand and embrace the complexity of our universe, and to continue to explore and discover new ways to understand and explain it.