Schwarzschild Metric - Need help understanding

In summary: Because that is what is needed to understand what these symbols represent. The "d" stands for distance and the "dr" stands for the derivative of distance with respect to time.Basically, the "d" is the distance between two points in space-time, and the "dr" is the change in that distance over time.
  • #1
Laserbeam
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Alright, first things first, I'm a grade 12 student residing in Ontario, Canada and I'm relatively new to these forums and to the world of physics.

I'm doing my grade 12 ISU and I've taught myself how to work around spherical coordinate systems, however, the schwarzschild metric confuses me.

I'm aware that the two angles represent the azimuthal angle and the angle of inclination (measured from the zenith), rs represents the schwarzschild radius, ts represents the "proper time" (measured in an inertial frame of reference), tm represents the time in motion, or the time measured by a distant observer for an event to occur, which is in a non-intertial frame of reference and of course c stands for the speed of light.

Where the problems begin are the d (distance) in the metric and the radii in the metric. I'm not sure what these stand for. Is the "r" the radial coordinate present in the spherical coordinate systems? To go along with this distance, I have no idea what that stands for to any degree in this metric. The distance to what?

This is all in relation to black holes, which is what I am doing my project on.

Thanks in advance!
 
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  • #2
Can you write out the Schwarzschild metric you are looking at? Your description of the "time" coordinates seems to be not the same as the usual t that appears in the metric. Usually there is only one t, and that is the coordinate time measured by an observer at infinity.

r in the usual Schwarzschild metric is the coordinate distance from the center. It is the same r as in the spherical coordinates.
 
  • #3
Laserbeam said:
Alright, first things first, I'm a grade 12 student residing in Ontario, Canada and I'm relatively new to these forums and to the world of physics.

I'm doing my grade 12 ISU and I've taught myself how to work around spherical coordinate systems, however, the schwarzschild metric confuses me.

I'm aware that the two angles represent the azimuthal angle and the angle of inclination (measured from the zenith), rs represents the schwarzschild radius, ts represents the "proper time" (measured in an inertial frame of reference), tm represents the time in motion, or the time measured by a distant observer for an event to occur, which is in a non-intertial frame of reference and of course c stands for the speed of light.

Where the problems begin are the d (distance) in the metric and the radii in the metric. I'm not sure what these stand for. Is the "r" the radial coordinate present in the spherical coordinate systems? To go along with this distance, I have no idea what that stands for to any degree in this metric. The distance to what?

This is all in relation to black holes, which is what I am doing my project on.

Thanks in advance!

Assuming I'm understanding you correctly (writing down the metric you're asking about would help to make sure there aren't any misunderstandings!) ts is not the "proper time", but is the Schwarzschild time coordinate.

ts in the Schwarzschild coordinate system is the time coordinate that an observer at infinity will assign to an event, (and it's not particularly well behaved :-( ).

r in the Schwarzschild metric is a radial coordinate. While all objects the same distance away from the black hole have the same value for r, r is not numerically equal to any sort of distance except by fortuitous accident.

Getting the distance between two points requires a couple of things. 1) It requires some calculus, I"m not sure if you have that. 2) It requires a knowledge of what the observer measuring the distance considers to be "simultaneous". In relativity, different observers have different notions of simultaneity.

To gloss over the process, though, coordinates are just values assigned to events in space-time, that are unique. They don't actually have any physical significance, they're like numbers on a map. The metric is what converts these coordinate values into distances, using calculus, when you have the additional needed knowledge of what events are considered to be simultaneous.

Have you had any special relativity background? If you have had some background, the relativity of simultaneity will be a familiar concept. if not, it may be new and hard to understand. However, it'll be hard to have a really detailed understanding of black holes without a good knowledge of special relativity, though you might be able to pick up a somewhat less-detailed understanding.

You might check out "Exploring Black Holes" by Taylor. It may or may not be too advanced, but the first few chapters are available online for free.

http://www.eftaylor.com/download.html#general_relativity
 
  • #4
If I understand correctly, you are asking what is the meaning of the "d" in the "dt" and "dr" and so on expressions. And for that you will need a bit more math than Canadian high school math. It is probably not too harmful if you thing of it as the symbol for difference so dt stands for t2-t1.
 
  • #5
The relationship between the Schwarzschild coordinate radius, r, and the "proper" radius, R, is discussed in section 11.5 of http://www.blau.itp.unibe.ch/lecturesGR.pdf . The meaning of the metric is that it gives the "proper" (ie. measured) lengths and times as a function of the coordinates. This is the case in flat and in curved spacetime. Information such as whether the spacetime is flat or curved comes from looking at derivatives of the metric.
 
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  • #7
Matterwave said:
Can you write out the Schwarzschild metric you are looking at? Your description of the "time" coordinates seems to be not the same as the usual t that appears in the metric. Usually there is only one t, and that is the coordinate time measured by an observer at infinity.

r in the usual Schwarzschild metric is the coordinate distance from the center. It is the same r as in the spherical coordinates.

Actually r isn't defined as the coordinate distance from the center in the metric in question. It is simply defined so that it gives the usual area and volume of a sphere that we are familiar with.
 
  • #8
Laserbeam said:
Is the "r" the radial coordinate present in the spherical coordinate systems? To go along with this distance, I have no idea what that stands for to any degree in this metric. The distance to what?
The r coordinate is not a distance except for a radially free falling observer falling at escape velocity (free falling from infinity). In all other cases one must apply the Lorentz transformation before integration to obtain the proper distance, which is usually represented by the letter rho.
 

FAQ: Schwarzschild Metric - Need help understanding

What is the Schwarzschild Metric?

The Schwarzschild Metric is a mathematical description of the curvature of space-time around a non-rotating, spherically symmetric mass. It is a solution to Einstein's field equations in general relativity, and it describes the gravitational field outside of a spherical object, such as a star or a black hole.

How does the Schwarzschild Metric relate to black holes?

The Schwarzschild Metric is often used to describe the gravitational field of a black hole. It predicts that at the event horizon of a black hole, the curvature of space-time becomes infinite, and anything that crosses the event horizon will be unable to escape the black hole's gravity.

What do the terms "singularity" and "event horizon" mean in the context of the Schwarzschild Metric?

In the Schwarzschild Metric, a "singularity" refers to a point of infinite density and curvature, which is located at the center of a black hole. The "event horizon" is the boundary around a black hole where the escape velocity is equal to the speed of light, making it impossible for anything to escape the black hole's gravitational pull.

How can the Schwarzschild Metric be used to understand the bending of light around massive objects?

According to the Schwarzschild Metric, the curvature of space-time around a massive object like a star or a black hole can cause light to follow a curved path. This phenomenon, known as gravitational lensing, has been observed and confirmed by astronomers, providing evidence for the validity of the Schwarzschild Metric and general relativity.

Can the Schwarzschild Metric be used to describe the entire universe?

The Schwarzschild Metric is a solution to Einstein's field equations that describes the gravitational field around a non-rotating, spherically symmetric mass. Therefore, it cannot be used to describe the entire universe, as the distribution of mass and energy in the universe is not spherically symmetric. Other metrics, such as the Friedmann-Lemaitre-Robertson-Walker metric, are used to describe the universe on a larger scale.

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