# Schwarzschild Metric - Need help understanding

#### Laserbeam

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.

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#### Matterwave

Gold Member
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.

#### pervect

Staff Emeritus
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.

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.

#### martinbn

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.

#### atyy

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 [Broken]. 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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#### WannabeNewton

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.

#### Passionflower

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.

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