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

In summary, 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.
  • #1
santo35
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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 ?) )
 
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  • #2
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).
 
  • #3
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.
 
  • #4
santo35 said:
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.
 
  • #5


Thank you for sharing your experience with exploring relativity and your pursuit of understanding it through a mathematical approach, specifically through Professor Susskind's lectures.

In regards to your question about the metric tensor, I can confirm that your initial understanding is correct. The metric tensor is not a constant quantity for all frames. It is a mathematical object that describes the relationship between coordinates in different frames, but it does not remain unchanged in different coordinates. As you mentioned, even vectors, which are scalars, can change when moving from one coordinate frame to another, so it makes sense that tensors, which are more complex mathematical objects, would also change.

The equation that relates the metric tensor in one frame to another is a way to mathematically describe this change and transformation between frames. It allows us to understand how measurements and observations can differ in different frames and how to reconcile these differences.

I encourage you to continue exploring and deepening your understanding of tensors and their role in relativity. They are a fundamental concept in the theory and are crucial to understanding the behavior of spacetime. Good luck in your studies!
 

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

What is a metric tensor?

A metric tensor is a mathematical object that is used in differential geometry to define the distance between points in a curved space. It is represented as a matrix and contains information about the geometry of the space.

How is a metric tensor related to frames?

A frame is a set of basis vectors that are used to describe a space. The metric tensor is used to define the inner product between these basis vectors, which determines the geometry of the space. In other words, the metric tensor tells us how to measure distances and angles in a given frame.

What is the significance of the metric tensor in physics?

The metric tensor is essential in physics, particularly in the theory of general relativity. It is used to describe the curvature of spacetime and is the mathematical basis for Einstein's field equations. It allows us to understand the effects of gravity on the motion of objects in the universe.

How is the metric tensor calculated?

The metric tensor is calculated using the line element in a given space. The line element is a differential expression that describes the distance between two points in a curved space. By integrating the line element, we can obtain the values for the metric tensor at each point in the space.

Can the metric tensor change in different frames?

Yes, the metric tensor can change in different frames. In general relativity, the metric tensor is not a fixed quantity, but it is dependent on the coordinates and frame in which it is being calculated. This is because the curvature of spacetime can vary in different frames, resulting in a different metric tensor.

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