Length and time for a given metric

In summary, the conversation discusses the topic of physical length and time in relation to the metric in general relativity. The speaker is looking for recommendations for a resource that explains these concepts at a basic level. They mention Landau's discussion, but find it confusing, particularly in regards to the concept of "proper time". The conversation also touches on the use of the metric to calculate distance in GR. The speaker expresses a desire to gain a better understanding of these concepts at an advanced level. They recommend chapter 9 of a specific book for further study.
  • #1
wandering.the.cosmos
22
0
I'm looking for recommendations for a good place that discusses at a basic level what physical length, time, simultaneity, etc. mean, for an arbitrary metric.

Landau does discuss this a bit, but in a way that confuses me -- for example he calls

[tex]\sqrt{g_{00}} dx^0[/tex]

"proper time", but in the SR limit this would just become dt, which isn't proper time, is it? Landau goes on to use this to derive distance in GR

[tex]\gamma_{ij} dx^i dx^j = \left( -g_{ij} + \frac{ g_{0i} g_{oj} }{g_{00}} \right) dx^i dx^j[/tex]

where my indices are spatial -- they run only from 1 to 3.

I'd like to understand where these expressions come from, and more importantly gain a good understanding of what length and time mean.
 
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  • #3


Length and time are fundamental concepts in physics, and they are closely related to each other. In the context of a given metric, length and time can be defined in terms of the metric components, which describe the geometry of the space-time in that particular metric. The metric components are usually denoted by g_{\mu\nu}, where \mu and \nu represent the four dimensions of space-time (three spatial dimensions and one time dimension).

In general relativity, the metric components are not constant and can change depending on the location and the gravitational field. This means that the concepts of length and time can also change depending on the metric. For example, in the special theory of relativity, the metric components are constant and the concepts of length and time are absolute. However, in general relativity, the metric components can vary and therefore the concepts of length and time are relative.

One way to understand the meaning of length and time in a given metric is to look at the geodesic equation. This equation describes the shortest path between two points in a given space-time. The length of this path is given by the metric components. Similarly, the time interval between two events can be defined in terms of the metric components. This is known as the proper time, which is the time measured by an observer who is at rest with respect to the space-time.

In the context of an arbitrary metric, the concepts of length and time can be quite complex. Landau's approach to defining these concepts may seem confusing at first, but it is based on the fundamental equations of general relativity. The expression \sqrt{g_{00}} dx^0 represents the proper time interval between two events, and it is derived from the geodesic equation. Similarly, the expression \gamma_{ij} dx^i dx^j represents the spatial distance between two points in a given metric.

It is important to note that in general relativity, the concepts of length and time are not absolute. They can vary depending on the observer and the gravitational field. This is why Landau's approach may seem confusing, as it takes into account the effects of gravity on the concepts of length and time.

In order to gain a better understanding of these concepts, it is recommended to study the fundamental equations of general relativity and the geodesic equation. This will help in understanding the meaning of length and time in a given metric and how they are related to each other. Additionally, studying the concept
 

What is a metric?

A metric is a standardized unit of measurement that is used to quantify and compare different quantities, such as length and time. It is a crucial component of scientific research and allows for accurate and consistent measurements.

How is length measured?

Length is typically measured using a metric unit, such as meters or centimeters. These units are based on the International System of Units (SI) and are defined by specific physical standards, such as the distance light travels in a vacuum in 1/299,792,458 of a second for the meter.

What is the relationship between length and time?

Length and time are two different quantities that can be measured using the same metric units. Length is a measure of distance, while time is a measure of duration. However, both length and time are closely related, as time is often used to measure the change in length, such as the speed of an object.

How do scientists use metrics to conduct experiments?

In scientific experiments, metrics are used to collect and analyze data in a standardized and precise manner. By using consistent metric units, scientists can compare and communicate their findings accurately, leading to a better understanding of the natural world.

Why is it important to use metric units in scientific research?

Metric units are important in scientific research because they provide a common language for scientists to communicate and share their findings. They also allow for precise and accurate measurements, making it easier to replicate experiments and validate results. Additionally, the use of metric units promotes international collaboration and facilitates the advancement of scientific knowledge.

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