Definition of a Tensor: Learn Physics Basics

[vikram_n]

I am still a physics novice and am learning new things everyday. I've been looking at tensors recently and I'm finding that I can't really understand what they are. Could someone explain in relatively simple words what the definition of a tensor is and why they are so significant? Also, what is the significance of Riemann's metric tensor, which I read about in Michio Kaku's Hyperspace. Examples would be greatly appreciated. Thanks!

 

[Mark M]

See this video:

http://www.youtube.com/watch?v=f5liqUk0ZTw

Take a look at this thread, where the OP asked a similar question. There are a variety of good explanations that you may find helpful.

https://www.physicsforums.com/showthread.php?t=623124

Also, you're mixing together two different tensors - the Riemann curvature tensor and the metric tensor.

 

[vikram_n]

Yes, my mistake. I meant the Riemann curvature tensor.

 

[Muphrid]

What is a tensor? Really, there are two kinds of tensors. One kind of tensor is a linear operator that, for example, maps vectors to other vectors. Hence the matrices representing rotations, reflections, and the like are representations of such tensors.

The other kind of tensor represents a generalization of a vector. From the span of two vectors, you can build a plane. From three vectors, you can build a volume, and so on. Some tensors represent these objects.

For both kinds of tensors, there is a transformation law based on the idea that a change of coordinates should not change the tensor itself--new coordinates may change the components, but the overall object should remain unchanged. It is this common transformation law (of the components) that is why both types of tensors--the linear operators and the generalization of vectors--are typically lumped together.

 

[pmacias]

A tensor is a mathematical object that represents the relationship between different physical quantities in a given system. It is a multidimensional array of numbers that transforms in a specific way under coordinate transformations. Tensors are important in physics because they allow us to describe the behavior of physical systems in a precise and consistent manner.

One of the key applications of tensors is in Einstein's theory of general relativity, where they are used to mathematically describe the curvature of spacetime. This is where Riemann's metric tensor comes into play. It is a specific type of tensor that characterizes the local curvature of spacetime at a given point. This is essential for understanding the effects of gravity and the motion of objects in the universe.

Another example of the significance of tensors can be seen in the study of fluid mechanics. Tensors are used to describe the stress and strain of fluids, which is important in understanding the behavior of fluids in various situations.

In short, tensors are essential mathematical tools in physics that allow us to describe and understand the behavior of physical systems. They have numerous applications in various fields of physics and are crucial for advancing our understanding of the universe.