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Covariants, Contra-variants, Invariants, Variants in SR?


TakenItSeriously
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I'm having a tough time getting a laypersons understanding for the differences between:

 

covariants, contravariants, invariants, and variants

 

as they relate to Lorentz transforms in the field of Special Relativity.

 

An intuitive definition for those terms would be wonderful, but I'd be extremely grateful even for some examples of each.

 

They don't need to be limited to the field of SR, but if outside of that field such as a references to GR instead, please be sure to list it as such.

 

Thanks.

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covariant and contravariant are directly related to the vector rotation direction in the Einstein field equation tensors. covariant is clockwise rotation. Or other tensors.

 

https://en.m.wikipedia.org/wiki/Covariance_and_contravariance_of_vectors

 

the term invariant means a measurement or equation that is the same measurement or mathematical relation for all observers

Edited by Mordred
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Simply, covariant and contravariant refers to tensor indices (upstairs index is contravariant, downstairs index is covariant). For a full understanding of what they mean, I recommend a textbook like Schutz's General Relativity textbook. He gives a great indroduction to tensors and the notation involved. Invariants are scalars which are the same in every reference frame. Variants (although I've never really seen this term in practice before) are simply scalars that change with reference frame.

 

EDIT: I realized examples would probably help.

 

Contravariant: An example of a contravariant tensor (rank-1, aka vector) would be velocity:

 

[math]v^\mu = \frac{dx^\mu}{d \tau}[/math]

 

An example of a covariant tensor would be the gradient of the electric potential [math]\phi[/math]:

 

[math]\partial_\mu \phi = \frac{\partial \phi}{\partial x^\mu}[/math].

 

One of the useful aspects of this notation is that summing over upper and lower indices of vectors produces invariants, i.e. objects that don't depend on your choice of reference frame:

 

[math]\sum_\mu q \, \partial_\mu \phi \, v^\mu = power[/math].

 

(Also, the electric potential was probably a bad example because it's actually part of the electromagnetic 4-potential. As such it will also transform under a coordinate transformation, so the above example isn't actually true. I chose it because it's the most familiar. Substitute it with a true scalar field for an accurate example.)

Edited by elfmotat
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Yes, only I realized I'm going to have to take some time studying up on tensors, and I haven't found the time yet.

 

Sorry, I didn't mean to forget to thank those who replied.

 

The terms contra and co variant were introduced in the middle of the 19 century during the study of coordinate systems and it is this meaning that you are seeking.

Be aware that mathematicians later generalised the concept to more abstract and general systems of algebra, called category theory and processes known as functors. Be glad you do not need these.

 

Vectors and tensors have elements and strictly speaking it is these elements that exhibit covariant, neutral or contravariant behaviour, not the vectors, tensors etc themselves.

 

So what is covariant, neutral or contravariant behaviour?

 

 

Well since many people are put off by the notation and complexity when it is introduced with tensors let us look at the simplest situation - that of a single component vector or tensor.

 

That's right you can have a single component vector or tensor say [a]

 

This is useful because one component can only exhibit one of the behaviours, whereas a vector or tensor of higher order can have some components contra, some neutral and some co variant.

 

OK so some concrete examples of this

 

I am some 5 miles away from the controversial nuclear power station at Hinkley point.

 

To put in a coordinate system, my map says that HP is 3 miles east and 4 miles north of me.

 

Now suppose that there has been a leak and some of radiation passes over such that the dose, R, radiation units is equal to

[math]R = \frac{{{\rm{Constant}}}}{{{\rm{Distance}}}} = \frac{a}{D}[/math]

But the formula is given out by the french operator, whose map measures distance in kilometers is R = 5/D. That is a = 5 for them.

 

To remedy the situation I need to modify their formula and I have two choices.

 

1) I can convert miles to kilometers

 

or

 

2) I can keep working in miles and change the constant to suit.

 

If I adopt (1) I will get a large number (8 kilometres) for D but the actual physical distance is the same.

But the actual units on my axes are smaller since kilometres are shorter than miles.

 

So R = 5/8 radiation units

 

This is an example of contravariance.

 

When I transform from miles to kilometers the variation in the size of the variable (larger) is in the opposite direction to the variation in the size of the units on the axes (smaller).

 

If I adopt (2) then the dose R stays the same and I must alter the dose factor a to suit - in this case multiply it by 5/8, making the equation

 

[math]R = \frac{5}{{D\,kilometrs}} = \frac{{5*\frac{5}{8}}}{{D\,miles}} = \frac{{25}}{{8D}}[/math]
Which again equals 5/8 radiation units.
But this time the variation (smaller in a) is the same as the variation in axis units (smaller).
This is an example of covariance.
A neutral component would be one that is unaffected by the transformation.
This is called an invariant.
So in the case of transforming a map in miles to one in kilometers bearing angles would remain the same.
If this helps we can move on to look more deeply into the subject.
Edited by studiot
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The terms contra and co variant were introduced in the middle of the 19 century during the study of coordinate systems and it is this meaning that you are seeking.

Be aware that mathematicians later generalised the concept to more abstract and general systems of algebra, called category theory and processes known as functors. Be glad you do not need these.

 

Vectors and tensors have elements and strictly speaking it is these elements that exhibit covariant, neutral or contravariant behaviour, not the vectors, tensors etc themselves.

 

So what is covariant, neutral or contravariant behaviour?

 

 

Well since many people are put off by the notation and complexity when it is introduced with tensors let us look at the simplest situation - that of a single component vector or tensor.

 

That's right you can have a single component vector or tensor say [a]

 

This is useful because one component can only exhibit one of the behaviours, whereas a vector or tensor of higher order can have some components contra, some neutral and some co variant.

 

OK so some concrete examples of this

 

I am some 5 miles away from the controversial nuclear power station at Hinkley point.

 

To put in a coordinate system, my map says that HP is 3 miles east and 4 miles north of me.

 

Now suppose that there has been a leak and some of radiation passes over such that the dose, R, radiation units is equal to

 

[math]R = \frac{{{\rm{Constant}}}}{{{\rm{Distance}}}} = \frac{a}{D}[/math]

 

But the formula is given out by the french operator, whose map measures distance in kilometers is R = 5/D. That is a = 5 for them.

 

To remedy the situation I need to modify their formula and I have two choices.

 

1) I can convert miles to kilometers

 

or

 

2) I can keep working in miles and change the constant to suit.

 

If I adopt (1) I will get a large number (8 kilometres) for D but the actual physical distance is the same.

But the actual units on my axes are smaller since kilometres are shorter than miles.

 

So R = 5/8 radiation units

 

This is an example of contravariance.

 

When I transform from miles to kilometers the variation in the size of the variable (larger) is in the opposite direction to the variation in the size of the units on the axes (smaller).

 

If I adopt (2) then the dose R stays the same and I must alter the dose factor a to suit - in this case multiply it by 5/8, making the equation

 

 

[math]R = \frac{5}{{D\,kilometrs}} = \frac{{5*\frac{5}{8}}}{{D\,miles}} = \frac{{25}}{{8D}}[/math]

 

Which again equals 5/8 radiation units.

 

But this time the variation (smaller in a) is the same as the variation in axis units (smaller).

 

This is an example of covariance.

 

 

A neutral component would be one that is unaffected by the transformation.

 

This is called an invariant.

 

So in the case of transforming a map in miles to one in kilometers bearing angles would remain the same.

 

If this helps we can move on to look more deeply into the subject.

Thanks a lot for this explanation. it answered my question.

 

i just wanted to fact check that my uses of the lorentz transform were properly handled and I wasnt familiar with those terms, as special cases other than invariant. It looks like I'm fine.

 

Thanks again

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OK, glad you liked it.

 

If you want to move on to examine the formulae Mordred and Elfmotat referred to then please tell us if you are familiar with the following

 

The difference between a function and the value of a function
These symbols
[math]\frac{{dy}}{{dx}}[/math]
and
[math]\sum\limits_{}^{} {} [/math]
Matrices
Column vectors
Row vectors
If need be I can include enough explanation with the next post.
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OK, glad you liked it.

 

If you want to move on to examine the formulae Mordred and Elfmotat referred to then please tell us if you are familiar with the following

 

The difference between a function and the value of a function

 

These symbols

 

[math]\frac{{dy}}{{dx}}[/math]

 

and

 

[math]\sum\limits_{}^{} {} [/math]

 

 

Matrices

 

Column vectors

 

Row vectors

 

If need be I can include enough explanation with the next post.

Thanks for the help I can answer all those correctly but I need to focus on something now. Thanks again!

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  • 7 months later...

The post by studiot is excellent. A more terse way of saying the same thing is to say that when you have two coordinate systems u and v and a transformation A relating them, some quantities get converted by by transforming them, and others get converted by *inverse* transforming them. That's the essence of the two categories. The examples that studiot gave is one example of each type. Specifically, when converting from miles to km, the number expressing a length will go up because it takes more than one km to make a mile, but numbers expressing flux per unit area will go down, because a square km is less area than a square mile.

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The post by studiot is excellent. A more terse way of saying the same thing is to say that when you have two coordinate systems u and v and a transformation A relating them, some quantities get converted by by transforming them, and others get converted by *inverse* transforming them.

This is not quite correct. The transformation [math]A[/math] that you refer to has of course a matrix representation. But if you mean to imply that the matrix for a contravariant transformation is the inverse of that for a covariant transformation, you'd be wrong - the 2 matrices are mutual transposes.

 

Now it is true that there are matrices/transformations where the transpose IS the inverse - these are called unitary transformations - in general coordinate transformations are not of this type. Specifically, the Lorentz transformation is not unitary.

 

Specifically, when converting from miles to km, the number expressing a length will go up because it takes more than one km to make a mile, but numbers expressing flux per unit area will go down, because a square km is less area than a square mile.

Maybe, but distance is a scalar quantity, and for any physically meaningful transformation, one requires that scalars be invariant. Doubly so in the case of distance, since again, physically meaningful transformations are compelled to preserve the metric.

 

The terms co- and contravariant, although out-moded, refer to vector-like entities ONLY

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