I know all of special relativity but I also know it is not enough to study GR. The mathematics is very complex. I have “Spacetime and Geometry” by S. Caroll. He has described it GR well enough but I do not know some of the maths signs and symbols he uses. So what are the prerequisites I need to have in maths. Is there a book that covers those math topics? Or is ther a book on GR that is simple enough for me to understand? Please help.

# [Physics] a suitable book to learn general relativity for high school students?

general-relativityresource-recommendationssoft-question

#### Related Solutions

The Physics work in this field is rigorous enough. Hawking and Ellis is a standard reference, and it is perfectly fine in terms of rigor.

### Digression on notation

If you have a tensor contraction of some sort of moderate complexity, for example:

$$ K_{rq} = F_{ij}^{kj} G_{prs}^i H^{sp}_{kq}$$

and you try to express it in an index-free notation, usually that means that you make some parenthesized expression which makes

$$ K = G(F,H)$$

Or maybe

$$ K = F(G,H) $$

Or something else. It is very easy to prove (rigorously) that there is no parentheses notation which reproduces tensor index contractions, because parentheses are parsed by a stack-language (context free grammar in Chomsky's classification) while indices cannot be parsed this way, because they include general graphs. The parentheses generate parse trees, and you always have exponentially many maximal trees inside any graph, so there is exponential redundancy in the notation.

This means that any attempt at an index free notation which uses parentheses, like mathematicians do, is bound to fail miserably: it will have exponentially many different expressions for the same tensor expression. In the mathematics literature, you often see tensor spaces defined in terms of maps, with many "natural isomorphisms" between different classes of maps. This reflects the awful match between functional notation and index notation.

### Diagrammatic Formalisms fix Exponential Growth

Because the parenthesized notation fails for tensors, and index contraction matches objects in pairs, there are many useful diagrammatic formalisms for tensorial objects. Diagrams represent contractions in a way that does not require a name for each index, because the diagram lines match up sockets to plugs with a line, without using a name.

For the Lorentz group and general relativity, Penrose introduced a diagrammatic index notation which is very useful. For the high spin representations of SU(2), and their Clebsch-Gordon and Wigner 6-j symbols, Penrose type diagrams are absolutely essential. Much of the recent literature on quantum groups and Jones polynomial, for example, is entirely dependent on Penrose notation for SU(2) indices, and sometimes SU(3).

Feynman diagrams are the most famous diagrammatic formalism, and these are also useful because the contraction structure of indices/propagators in a quantum field theory expression leads to exponential growth and non-obvious symmetries. Feynman diagrams took over from Schwinger style algebraic expressions because the algebraic expressions have the same exponential redundancy compared to the diagrams.

Within the field of theoretical biology, the same problem of exponential notation blow-up occurs. Protein interaction diagrams are exponentially redundant in Petri-net notation, or in terms of algebraic expressions. The diagrammatic notations introduced there solve the problem completely, and give a good match between the diagrammatic expression and the protein function in a model.

Within the field of semantics within philosophy (if there is anything left of it), the ideas of Frege also lead to an exponential growth of the same type. Frege considered a sentence as a composition of subject and predicate, and considered the predicate a function from the subject to meaning. The function is defined by attaching the predicate to the subject. So that "John is running" is thought of as the function "Is running"("John").

Then an adverb is a function from predicates to predicates, so "John is running quickly" means ("quickly"("Is running"))("John"), where the quickly acts on "is running" to make a new predicate, and this is applied to "John".

But now, what about adverb modifiers, like "very", as in "John is running very quickly"? You can represent these are functions from adverbs to adverbs, or as functions from predicates to predicates, depending on how you parenthesize:

(("very"("quickly"))("Is running"))("John")

vs.

(("very")(("quickly")("Is running"))("John")

Which of these two parenthetization is correct define two schools of semantic philosophy. There is endless debate on the proper Fregian representation of different parts of speech. The resolution, as always, is to identify the proper diagrammatic form, which removes the exponential ambiguity of parenthesized functional representation. The fact that philosophers have not done this in *100 years* of this type of debate on Fregian semantics shows that the field is not healthy.

I believe that the geometric point of view is superior to the algebraic one in quantum theory. Many of the achievements in understanding quantum theory emerged from the geometrical point of view, for example, Wigner's classification of relativistic particles (as irreducible representations of the Poincare group). Also, many of Witten's achievements stemmed from his deep geometrical understanding. In fact, in his seminal works he applied geometric quantization beyond the limits that were known to mathematicians at the time.

Of course, the mathematical areas relevant to this direction of research include: Analysis on manifolds, Lie groups, Fibre bundles, Symplectic geometry, Geometric quantization Etc.

In the special case of QIT, it is true that the main stream follows the algebraic point of view, but let me refer you to works adopting the geometric point of view. The basic reference is Bengtsson and Zyczkowski's book: Geometry of quantum states: An introduction to quantum entanglement. Let me also refer you to important more recent works in this direction:

Geometry of entangled states by Marek Kus and Karol Zyczkowski.

Symplectic geometry of entanglement by: Adam Sawicki, Alan Huckleberry, Marek Kus, and

Segre maps and entanglement for multipartite systems of indistinguishable particles by: Janusz Grabowski, Marek Kus, Giuseppe Marmo

These articles include many other references on the subject, also, many of the authors have additional works.

## Best Answer

High school student here. I've tried reading Carroll as well, and I also found the math to be quite difficult. However, the good news is that not all of the math he introduces is necessary to understand the subject. It is possible to develop an intuitive understanding of GR without formal topology and differential geometry. For this, I recommend Hartle: https://www.amazon.com/Gravity-Introduction-Einsteins-General-Relativity/dp/0805386629. It's much more self-contained and accessible than Carroll. I learned about this book after I spent a year struggling through Carroll to learn GR, but I would have fared much better if I had started with this.