<p><em>ds</em> refers to a distance in spacetime (this includes time). Just as, in classical physics, you see <span class="math-container">$$ds^2=dx^2+dy^2+dz^2$$</span> for any given line element, in spacetime you get similar things going on. For flat spacetime, this gives some good intuitive understanding. Depending on your signature, an element in flat (Minkowski) spacetime is <span class="math-container">$$ds^2=dt^2-dx^2-dy^2-dz^2$$</span> or <span class="math-container">$$ds^2=-dt^2+dx^2+dy^2+dz^2$$</span> Since the whole point of general relativity is the curvature of spacetime, a line element in curved space will be different. So for the Schwarzschild metric we see what you're seeing in the pictures you provided, except instead of cartesian spatial coordinates, spherical coordinates are used and since the Schwarzschild metric is non-rotating and spherically symmetric, we do not see any angular elements in the metric.</p>

CC BY-SA 4.0
2023-01-10T12:05:41.887Z
great explanation. :)
William Martens
CC BY-SA 4.0
2021-11-08T22:47:36.807Z
2021-11-08T22:47:36.807Z
Justin T
<p><a href="https://i.sstatic.net/QVShh.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/QVShh.png" alt="enter image description here" /></a></p>
<p>And:</p>
<p><a href="https://i.sstatic.net/eyk0h.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/eyk0h.png" alt="enter image description here" /></a></p>
<p><a href="https://i.sstatic.net/VUjRG.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/VUjRG.png" alt="enter image description here" /></a></p>
<p><a href="https://i.sstatic.net/iqsNH.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/iqsNH.png" alt="enter image description here" /></a></p>
<p>These are from Wikipedia on Schwarzschild metric, Derivation of the Schwarzschild metric and the last is from Science Direct, Schwarzschild metric.</p>
<p><img src="https://i.sstatic.net/8VIEE.png" alt="enter image description here" /></p>

CC BY-SA 4.0
2021-11-08T22:37:39.940Z
What are the black bars meant to be.
James K
CC BY-SA 4.0
2021-11-09T06:32:37.027Z
Forget the black bars.... I don't know why copying and pasting images from Wikipedia didn't work.... It works for text.... And I wasn't warned by any message or software that it would be blacked out....
Kurt Hikes
CC BY-SA 4.0
2021-11-09T11:12:47.853Z
You might actually *link* to the text. And quote the interesting parts as actual text.
planetmaker
CC BY-SA 4.0
2021-11-09T11:40:33.360Z
I think that sometimes if the png has an alpha channel the conversion to SE's pngs does this. If you can convert to jpeg first somehow (which does not have the alpha channel) that might fix it.
uhoh
CC BY-SA 4.0
2023-01-09T16:33:35.280Z
[related and links therein(black holes and links explain also)](https://physics.stackexchange.com/q/735441/276316)
William Martens
<black-hole><history><general-relativity><theories>
What does 's' or 'ds' or $ds^2$ mean in the Schwarzschild equation(s)?
Kurt Hikes
# What does 's' or 'ds' or $ds^2$ mean in the Schwarzschild equation(s)?

(Asked by: username_0 on 2021-11-08)

[![enter image description here](https://i.sstatic.net/QVShh.png)](https://i.sstatic.net/QVShh.png)

And:

[![enter image description here](https://i.sstatic.net/eyk0h.png)](https://i.sstatic.net/eyk0h.png)

[![enter image description here](https://i.sstatic.net/VUjRG.png)](https://i.sstatic.net/VUjRG.png)

[![enter image description here](https://i.sstatic.net/iqsNH.png)](https://i.sstatic.net/iqsNH.png)

These are from Wikipedia on Schwarzschild metric, Derivation of the Schwarzschild metric and the last is from Science Direct, Schwarzschild metric.

![enter image description here](https://i.sstatic.net/8VIEE.png)

**Tags:** black-hole, history, general-relativity, theories

> **Comments:**
> **username_1**: What are the black bars meant to be.  
> **username_0**: Forget the black bars.... I don't know why copying and pasting images from Wikipedia didn't work.... It works for text.... And I wasn't warned by any message or software that it would be blacked out....  
> **username_2**: You might actually *link* to the text. And quote the interesting parts as actual text.  
> **username_3**: I think that sometimes if the png has an alpha channel the conversion to SE's pngs does this. If you can convert to jpeg first somehow (which does not have the alpha channel) that might fix it.  
> **username_4**: [related and links therein(black holes and links explain also)](https://physics.stackexchange.com/q/735441/276316)  

---

## 1 answers

### Answer by username_5 on 2021-11-08. Score: 4 | Accepted answer

*ds* refers to a distance in spacetime (this includes time). Just as, in classical physics, you see $$ds^2=dx^2+dy^2+dz^2$$ for any given line element, in spacetime you get similar things going on. For flat spacetime, this gives some good intuitive understanding. Depending on your signature, an element in flat (Minkowski) spacetime is $$ds^2=dt^2-dx^2-dy^2-dz^2$$ or $$ds^2=-dt^2+dx^2+dy^2+dz^2$$ Since the whole point of general relativity is the curvature of spacetime, a line element in curved space will be different. So for the Schwarzschild metric we see what you're seeing in the pictures you provided, except instead of cartesian spatial coordinates, spherical coordinates are used and since the Schwarzschild metric is non-rotating and spherically symmetric, we do not see any angular elements in the metric.

> **Comments:**
> **username_4**: great explanation. :)  

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