Difference between covariant and contravariant tensors
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Let's work in the three dimensions of classical space (forget time, relativity, four-vectors etc). Then the prototypical example of a contravariant vector is a displacement vector. Suppose you have a vector (1, 2, 3) representing a displacement. Implicitly it has units of meters or the like for each of the three components. Now suppose you shrink the coordinate system in the x direction, so that you're now measuring x in millimeters. This isn't something you'd do for the fun of it, but it's representative of things you might be forced to do when trying to lay out a coordinate system in curved space. (Think of lines of latitude and longitude on the surface of the earth - they're not, and can't be, equally spaced.)
To represent the same physical displacement, you have transform the vector to (1000, 2, 3). Implicitly that's (1000 mm, 2 m, 3 m). The vector component has gotten bigger as the coordinate system length unit has gotten smaller. That's contravariance.
Now imagine that you have a potential in that 3D space, and a gradient vector that tells in which direction the potential is increasing most rapidly and what that rate of increase is. That's what electric field is telling you about voltage. Implicitly the gradient vector has units of something-or-other per meter or the like - volts per meter in the case of electric field. So if the electric field was (4, 5, 6) V/m in the original coordinate system, it has to be (0.004 V/mm, 5 V/m, 6 V/m) in the coordinate system with shrunken x. The vector component has varied in the same way as the coordinate system length unit - that's covariance.
Now in classical physics you never have to think about this distinction because you're guaranteed to be able to use nice, simple, straight Euclidean coordinate systems with equal scales in all directions. You can get a taste by thinking about coordinate systems on the surface of a sphere, but there's not really a practical necessity.
However in GR, you get your nose rubbed in it because spacetime itself is modelled as bent, so there are no totally straight coordinate systems. So while you can continue to think of contravariant vectors as spear-like objects as usual, a better metaphor for the covariant vectors is as a stack of equipotential surfaces of whatever the implied potential is, as in the pictures at Covariance and contravariance of vectors.13.4k Views · 46 UpvotesDid you like this answer?Upvotes help us know what content you like.Upvote46DownvoteShareAsk Follow-UpRecommendedAll
Promoted by HT MediaAccenture, Cisco and JP Morgan are hiring IT professionals.Send in your application to get hired. Competitive salary. Apply now.Apply now at shine.comJoseph Wang, Ph.D. AstrophysicsAnswered May 20, 2014Originally Answered: What is the difference between a covariant derivative and a contravariant derivative of a tensor?I've always found it easier to think in terms of dimensional analysis.
Imagine a car travelling in a highway. The units of measure for the velocity of the car is going to be (length) / (time).
Now imagine that as this car is travelling, you what to measure how the temperature changes as the car goes in different directions. The temperature is going to be in centigrade, and the vector that describes how the temperature changes as you move in different directions is going to be (temperature) / (length).
You'll notice that in the first situation, the length is at the top, and in the second, the length is at the bottom. This means that if you do a unit conversion (say from feet to meters), you will end up doing the conversion in different ways. A contravariant vector is the name for the first type of vector, whereas a covariant vector is the name for the second type.
To represent the same physical displacement, you have transform the vector to (1000, 2, 3). Implicitly that's (1000 mm, 2 m, 3 m). The vector component has gotten bigger as the coordinate system length unit has gotten smaller. That's contravariance.
Now imagine that you have a potential in that 3D space, and a gradient vector that tells in which direction the potential is increasing most rapidly and what that rate of increase is. That's what electric field is telling you about voltage. Implicitly the gradient vector has units of something-or-other per meter or the like - volts per meter in the case of electric field. So if the electric field was (4, 5, 6) V/m in the original coordinate system, it has to be (0.004 V/mm, 5 V/m, 6 V/m) in the coordinate system with shrunken x. The vector component has varied in the same way as the coordinate system length unit - that's covariance.
Now in classical physics you never have to think about this distinction because you're guaranteed to be able to use nice, simple, straight Euclidean coordinate systems with equal scales in all directions. You can get a taste by thinking about coordinate systems on the surface of a sphere, but there's not really a practical necessity.
However in GR, you get your nose rubbed in it because spacetime itself is modelled as bent, so there are no totally straight coordinate systems. So while you can continue to think of contravariant vectors as spear-like objects as usual, a better metaphor for the covariant vectors is as a stack of equipotential surfaces of whatever the implied potential is, as in the pictures at Covariance and contravariance of vectors.13.4k Views · 46 UpvotesDid you like this answer?Upvotes help us know what content you like.Upvote46DownvoteShareAsk Follow-UpRecommendedAll
Promoted by HT MediaAccenture, Cisco and JP Morgan are hiring IT professionals.Send in your application to get hired. Competitive salary. Apply now.Apply now at shine.comJoseph Wang, Ph.D. AstrophysicsAnswered May 20, 2014Originally Answered: What is the difference between a covariant derivative and a contravariant derivative of a tensor?I've always found it easier to think in terms of dimensional analysis.
Imagine a car travelling in a highway. The units of measure for the velocity of the car is going to be (length) / (time).
Now imagine that as this car is travelling, you what to measure how the temperature changes as the car goes in different directions. The temperature is going to be in centigrade, and the vector that describes how the temperature changes as you move in different directions is going to be (temperature) / (length).
You'll notice that in the first situation, the length is at the top, and in the second, the length is at the bottom. This means that if you do a unit conversion (say from feet to meters), you will end up doing the conversion in different ways. A contravariant vector is the name for the first type of vector, whereas a covariant vector is the name for the second type.
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