Metric Tensor and frames (wrt prof.susskind's lectures)

[santo35]

my exploration of relativity followed by first reading various books which failed to explain to me how relativity worked but built a strong feel of how one can think about it. after which i decided to take the mathematical way of understanding it for which i am going on with the prof susskind's lecture

a quick doubt in it , in allmost all the first few parts of the series he says that tensor is a quantity that remains unchanged in different coordinates (is it ?) but derives the equation relating metric tensor in one frame and another coordinate frame. my understanding of tensor isn't that much strong.



PS: i think the first statement i made is wrong, as far as i conceive tensors i guess they are not constant for all frames.( as vector is a scalar and even it does change when moving from one flat coordinate to another curved surface coordinate (again does it ?) )

 

[elfmotat]

A tensor is just a generalization of a vector. Its components change from one coordinate system (frame) to another, but the vector itself doesn't (i.e. its magnitude - what it "physically represents" is the same in all frames).

 

[Chestermiller]

A vector can be represented as the summation of (scalar) components times unit vectors in the coordinate directions. If you change coordinate systems, both the components of the vector and the unit vectors for the new coordinate system are different from those for the old coordinate system, but the sum of the components times the unit vectors are the same for both coordinate systems. In the case of a second order tensor, we can represent it as the summation of (scalar) components times combinations of the coordinate unit vectors taken two at a time (so called dyadics). This summation does not change when you change coordinate systems.

 

[Dale]

in allmost all the first few parts of the series he says that tensor is a quantity that remains unchanged in different coordinates (is it ?) but derives the equation relating metric tensor in one frame and another coordinate frame.

A tensor is a geometric quantity that remains unchanged in different coordinates.

Tensors can be represented in terms of a set of basis vectors at each point. When you do that you can talk about the components of the tensor in that basis. It is important to make a mental distinction between the tensor and the components of the tensor in a given basis.

There are an infinite number of possible basis sets. The most common basis is the so-called coordinate basis. When you talk about tensors changing in different coordinates what you are really talking about are the components of the tensors under the coordinate basis of the different coordinate system. The geometric object is not changing, but the coordinate basis is, and therefore the components.