When are two spacetimes the same

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If you want to compare two spacetimes, you need to compare both the metrics and the coordinate charts. In summary, comparing metrics and coordinate charts is necessary to determine if two spacetimes are the same, even if they are described using different coordinates.
  • #1
smallphi
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Suppose you have two spacetimes described by two metrics.

If I prove that they have foliations with the same geometrical properties, would that prove the spacetimes are actually the same only in different coordinates?

Has anyone seen a source with theorems of this type?
 
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  • #2
smallphi said:
Suppose you have two spacetimes described by two metrics.

If I prove that they have foliations with the same geometrical properties, would that prove the spacetimes are actually the same only in different coordinates?

Has anyone seen a source with theorems of this type?
The first thing to distinguish is a metric from a coordinate chart. Different coordinate charts can refer to the same metric.
 
  • #3


The concept of whether two spacetimes are the same is a complex and debated topic in the field of physics. It is important to note that there is no definitive answer to this question, as it ultimately depends on one's interpretation and understanding of spacetime.

In general, two spacetimes can be considered the same if they have the same physical properties and allow for the same laws of physics to be applied. This means that even if they are described by different metrics, as long as these metrics result in the same physical predictions, the spacetimes can be considered equivalent.

However, proving that two spacetimes are the same is not a simple task. One approach, as mentioned in the content, is to show that they have foliations with the same geometrical properties. This means that the spacetimes can be divided into smaller regions that have the same geometric structure. This can be a valid way to show that the spacetimes have similar properties, but it does not necessarily prove that they are the same.

Another approach is to show that the two spacetimes can be transformed into each other through a continuous transformation, such as a coordinate transformation. This would mean that the two spacetimes are just different representations of the same underlying structure.

It is also worth noting that in certain cases, two spacetimes can be considered the same even if they have different metrics. This is known as diffeomorphism invariance, which is a fundamental principle in general relativity. It states that the laws of physics should be independent of the coordinate system used to describe them.

In terms of sources for theorems on when two spacetimes are the same, there is no one definitive source as this is an ongoing area of research. However, some relevant literature on the topic includes the work of mathematician Hermann Weyl, who introduced the concept of conformal invariance in spacetime, and the work of physicist Roger Penrose, who has proposed various criteria for identifying when two spacetimes are equivalent.

In conclusion, determining when two spacetimes are the same is a complex and ongoing topic in physics, and there is no one definitive answer or source for theorems on this topic. It ultimately depends on one's interpretation and understanding of spacetime and the criteria used to define equivalence.
 

1. What is a spacetime?

A spacetime is a mathematical model that combines the three dimensions of space and the dimension of time into a single four-dimensional continuum. It is used to describe the physical universe and the way objects move and interact within it.

2. How can two spacetimes be considered the same?

Two spacetimes are considered the same if they have the same mathematical structure and describe the same physical reality. This means that the metric, which defines the distances and intervals between points in spacetime, is the same for both spacetimes.

3. What are some examples of two identical spacetimes?

Some examples of identical spacetimes include the spacetime of a stationary object in a uniform gravitational field and the spacetime of an object moving at a constant velocity in a vacuum. Both of these spacetimes are described by the same metric, known as the Minkowski metric.

4. Can two different observers experience the same spacetime differently?

Yes, two different observers can experience the same spacetime differently depending on their relative motion and position in spacetime. This is known as the observer's frame of reference and is described by the theory of relativity.

5. Why is it important to determine when two spacetimes are the same?

Determining when two spacetimes are the same is important for understanding the fundamental laws of physics and how they apply in different situations. It also allows for the prediction and explanation of physical phenomena, such as gravitational waves and the behavior of objects in extreme conditions like black holes.

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