In this post, we will delve deeper into the concepts of differential geometry, mainly focusing on the notion of riemann tensor, ricci tensor, and ricci scalar.
I. The Riemann Tensor
A. Definition
We start with a \(d\)-dimensional manifold \(M\) endowed with a (pseudo-)Riemannian metric \(g_{\mu\nu}\). The connection compatible with the metric is the Levi-Civita connection \(\nabla\). The Riemann curvature tensor (often called simply the Riemann tensor) \(R^\rho_{\ \sigma\mu\nu}\) is defined by the commutator of covariant derivatives acting on a vector field \(V^\rho\):
In this post, I would like to build some basics of differential geometry for understanding the Einstein Field Equations.
1. Riemann Tensor and Its Properties
Definition of the Riemann Tensor:
The Riemann curvature tensor \( R^i_{jkl} \) is a fundamental object in differential geometry, representing the intrinsic curvature of a manifold. It is defined as:
\[ R^i_{jkl} = \partial_k \Gamma^i_{jl} - \partial_l \Gamma^i_{jk} + \Gamma^i_{km} \Gamma^m_{jl} - \Gamma^i_{lm} \Gamma^m_{jk} \]where \( \Gamma^i_{jk} \) are the Christoffel symbols.
I would like to learn some basic information about treasury bonds. The following questions are of interest to me:
- How the coupon rate is determined?
- How the yield is determined?
- What’s the relationship between yield and price?
- The relationship between yield and interest rate?
- From a macroeconomic perspective, what’s the status of the treasury bond market of USA, China, and Japan?
We refer to these lectures.
Treasury Bonds are debt securities issued by the government to finance its operations. They are known for their stability and are key instruments in the financial markets. Below is a structured explanation of the key concepts related to Treasury Bonds.
In this post, I would like to build some fundational knowledge about the Einstein Field Equations. First, I will pose several questions based on the last post about cosmology:
- What are the einstein tensor and the stress-energy tensor?
- How is Einstein Field Equations derived?
- In the context of cosmology, what is the perfect fluid approximation?
- How can we derive the Friedmann Equation and Acceleration Equation from the Einstein Field Equations?
We refer to this course.
I’m trying to learn financial investment, so I will first attempt to read financial reports to understand the business activities happening in the world.
Here are some fundamentals to get know about a company:
- Income Statement (Profit & Loss): Shows the company’s revenues, expenses, and profits over a specific period. Key metrics to focus on: revenue, gross profit, operating profit, net income. It’s kind of a video shows the activity of the company in a past period.
- Balance Sheet: Provides a snapshot of the company’s assets, liabilities, and equity at a specific point in time. It gives insight into the company’s financial health and how it’s funded. It’s like a snapshot.
- Cash Flow Statement: Highlights the company’s cash inflows and outflows over a period. It focuses on operating, investing, and financing activities. A key metric here is free cash flow, which indicates the amount of cash a company has left after capital expenditures.
- Statement of Shareholders’ Equity: Explains changes in equity from the previous period.
Difference of Income Statement and Cash Flow Statement
1. Income Statement (Profit & Loss Statement)
What it focuses on:
The Income Statement shows a company’s revenues and expenses over a specific period (e.g., a quarter or a year), and ultimately calculates the net income (profit or loss). It focuses on accrual accounting, which means it records revenues when earned and expenses when incurred, regardless of when the actual cash is received or paid.
I want to learn about cosmology, maybe start with the book “An Introduction to Modern Cosmology” by Andrew Liddle and the lecture “The Early Universe” by MIT OpenCourseWare.
The standard Big Bang
It does not tell the cause of the Bang, but the aftermath of the Bang. It assumes all the matter already existed before the Bang.
Cosmic Inflation
A prequal to the Big Bang.
- gravity can be repulsive when pressure is negative. (Find out more about the relationship between gravity and pressure in the next part.)
There is a patch of repulsive gravity material in the early universe, which is the cause of the Big Bang.
The (mass/energy) density of the repulsive material is not lowered as it expands.The resolution is to introduce negative energy.
Complementarity: The relationship between gravity and pressure
Below is a high-level introduction to these ideas, followed by some of the key formulas in cosmology and general relativity that show why negative pressure can produce a repulsive gravitational effect.