<p>The other question was closed, so I put this here.  One should watch these two PBS Space Time episodes that explain that mass causes time dilation, which is a curvature of space time, and with just the altered <span class="math-container">$g_{tt}$</span> metric one has gravitational effects.  Time dilation and reduced  <span class="math-container">$g_{tt}$</span> metric are nearly synonymous.  In the second episode, he discusses that if light is a wave then gravity bends its path by time dilation, just as refraction bends light by changing its speed.  But, if light is a particle one needs to see it as a curvature of space and time.</p>
<p><a href="https://www.pbs.org/video/does-time-cause-gravity-00kr0q/" rel="nofollow noreferrer">Does time cause gravity</a></p>
<p><a href="https://www.pbs.org/video/how-does-gravity-affect-light-zx47ji/" rel="nofollow noreferrer">How does gravity affect light</a></p>
<p>The motion of a particle in a weak static field can be found by starting with the geodesic equation of motion, a general solution of Einstein's field equations:</p>
<p><span class="math-container">$$
\frac{\mathrm{d}^2 x^\alpha}{\mathrm{d}\tau^2} + \Gamma^\alpha_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}\tau}\frac{\mathrm{d}x^\nu}{\mathrm{d}\tau} = 0
$$</span>
where <span class="math-container">$x^\alpha$</span>
is the 4-vector describing the position of a test particle and <span class="math-container">$\tau$</span> is proper time.</p>
<p>When the motion is far below from relativistic speeds, we can neglect <span class="math-container">$\frac{\mathrm{d}x^i}{\mathrm{d}\tau} (i=1..3$</span>) compared to the <span class="math-container">$i=0$</span> component (<span class="math-container">$t=x^0$</span>):</p>
<p><span class="math-container">$$
\frac{\mathrm{d}^2 x^\alpha}{\mathrm{d}\tau^2} + \Gamma^\alpha_{00}\big(\frac{\mathrm{d}t}{\mathrm{d}\tau}\big)^2=0.
$$</span></p>
<p>In a stationary gravitational field, all time derivatives vanish, thus only one of the three terms of the connection coefficients survives:</p>
<p><span class="math-container">$$
 \Gamma^\alpha_{00} = -\frac{1}{2} g^{\mu\nu} \frac{\partial{g_{00}}}{\partial{x^v}}.
$$</span></p>
<p>For weak gravitational fields the metric can be written as small perturbations, <span class="math-container">$h$</span>, to the Minkowski metrics, <span class="math-container">$\eta$</span>,</p>
<p><span class="math-container">$$
g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}.
$$</span></p>
<p>Ignoring terms of second order in the <span class="math-container">$h_{\mu\nu}$</span>, we have
<span class="math-container">$$
\Gamma^\alpha_{00} =  -\frac{1}{2} \eta^{\mu\nu} \frac{\partial{h_{00}}}{\partial{x^\nu}},
$$</span>
which is just the gradient of the perturbation to time.</p>
<p>From this it is evident that the orbit in the weak field depends only on the perturbation to time, specifically <span class="math-container">$h_{00}$</span> and <span class="math-container">$\frac{\mathrm{d}t}{\mathrm{d}\tau}$</span>, the time dilation.  And it is this which implies that such orbits depend only on curvatures in time dimension of 4 space.</p>
<p>Continuing on, if one sets <span class="math-container">$h_{00} = 2 \phi$</span>, one finds the geodesic equation is</p>
<p><span class="math-container">$$
\frac{\mathrm{d}^2 \textbf{x}}{\mathrm{d}t^2} = - \nabla \phi
$$</span></p>
<p>which is Newtonian law.  To express this as plainly as possible:
<span class="math-container">$$
\ddot{\textbf{x}} = - \frac{1}{2}\nabla h_{00}
$$</span>
which says acceleration is equal to minus one-half the gradient of the perturbation of the time metric.</p>
<p>Much of this is presented in the chapter &quot;Newtonian Gravity in the Language of Curved Spacetime&quot; in &quot;Gravitation&quot; by Misnor, Thorne, and Wheeler.</p>

CC BY-SA 4.0
2024-12-25T05:33:39.597Z
Mite, I have seen 'em all, on Youtube, and I am even more confused, "MK"? I thought I understood GR and gravity, until this claim that gravity is really caused by GTD, started appearing, all over the place. I can see how GTD can deflect photons but how about matter/objects? Wouldn't GTD just make my feet a bit younger, than my head? If GTD is a measure of time curvature, then what is a simple measure of space curvature?
Blankety Blank
CC BY-SA 4.0
2024-12-25T14:27:22.987Z
You are traveling through 4-space.  If your feet are traveling more slowly then you rotate in 4-space.  Your geodesic now points toward the gravity source.   
    We don’t have a distinct word for space curvature.  For non-relativistic motion c$\Delta t >> \Delta x$ so effects of space curvature are insignificant.   Curvature appears flat locally.  Precession of Mercury is affected though.
eshaya
CC BY-SA 4.0
2024-12-25T04:05:12.890Z
2024-12-28T19:15:09.157Z
2024-12-28T19:15:09.157Z
eshaya
<p>In this footnote on P14 in his 2011 book &quot;The Hidden Reality&quot;?</p>
<blockquote>
<p>It's easier to envision curved space than curved time, and that's why many popularizations of Einsteinian gravity focus solely on the former. However, for the gravity generated by familiar objects like the Earth and Sun, it is actually the curvature of time - not space - that exerts the dominant impact. For an illustration, think of 2 clocks, one on the ground, the other on top of Empire State Building. Because the ground clock is closer to the Earth's centre, it experiences slightly stronger gravity than the clock that's high above Manhattan. Generel relativity/GR shows that because of this, the rate at which time passes on each will be slightly different: the ground clock will run a tiny bit slow (billionths of a sec/ond / year) compared to the elevated clock. The temporal mismatch is an example of what we mean by time being curved or warped. GR then establishes that objects move towards regions where time elapses more slowly; in a sense, all objects &quot;want&quot; to age as slowly as possible. From an Einstenian perspective, that explains why an object falls when you let go of it.</p>
</blockquote>
<p>Is he saying that gravity really is caused in part, at least, by gravitational time dilation, as many others do?!</p>

CC BY-SA 4.0
2024-12-24T17:34:15.280Z
This is essentially the same question as before, with the same answer.  Dilation is a consequence of space-time curvature, and not a cause of it.
James K
CC BY-SA 4.0
2024-12-25T05:24:06.493Z
Yes, matter warps/"curves" space, and?time, around, itself, which then causes GTD, and/but he/they then say that that then causes gravity! How exactly? I can see how GTD can deflect photons but how about matter/objects? Wouldn't GTD just make my feet a bit younger, than my head?
Blankety Blank
CC BY-SA 4.0
2024-12-25T09:58:41.783Z
Did you understand [my comment](https://astronomy.stackexchange.com/questions/59087/is-gravity-really-caused-by-gravitational-time-dilation#comment137198_59097) on the previous question? $g_{00}$ is the purely time component of the spacetime metric matrix. (The convention is to index time as the zeroth dimension, not the fourth).
PM 2Ring
CC BY-SA 4.0
2024-12-25T10:12:11.217Z
It's tricky getting an intuitive feel for this stuff. Our intuitions evolved to deal with speeds we normally encounter, eg the duration of a heartbeat feels comparable to the length of a pace, so 1 second ~ 1 metre. But in relativity, the natural speed is *c*, and it's hard to appreciate that 1 millisecond is the same magnitude as 300 km.
PM 2Ring
<gravity><general-relativity><space-time><time-dilation>
What is Brian Greene saying here about gravity?
Blankety Blank
# What is Brian Greene saying here about gravity?

(Asked by: username_0 on 2024-12-24)

In this footnote on P14 in his 2011 book "The Hidden Reality"?

> It's easier to envision curved space than curved time, and that's why many popularizations of Einsteinian gravity focus solely on the former. However, for the gravity generated by familiar objects like the Earth and Sun, it is actually the curvature of time - not space - that exerts the dominant impact. For an illustration, think of 2 clocks, one on the ground, the other on top of Empire State Building. Because the ground clock is closer to the Earth's centre, it experiences slightly stronger gravity than the clock that's high above Manhattan. Generel relativity/GR shows that because of this, the rate at which time passes on each will be slightly different: the ground clock will run a tiny bit slow (billionths of a sec/ond / year) compared to the elevated clock. The temporal mismatch is an example of what we mean by time being curved or warped. GR then establishes that objects move towards regions where time elapses more slowly; in a sense, all objects "want" to age as slowly as possible. From an Einstenian perspective, that explains why an object falls when you let go of it.

Is he saying that gravity really is caused in part, at least, by gravitational time dilation, as many others do?!

**Tags:** gravity, general-relativity, space-time, time-dilation

> **Comments:**
> **username_1**: This is essentially the same question as before, with the same answer.  Dilation is a consequence of space-time curvature, and not a cause of it.  
> **username_0**: Yes, matter warps/"curves" space, and?time, around, itself, which then causes GTD, and/but he/they then say that that then causes gravity! How exactly? I can see how GTD can deflect photons but how about matter/objects? Wouldn't GTD just make my feet a bit younger, than my head?  
> **username_2**: Did you understand [my comment](https://astronomy.stackexchange.com/questions/59087/is-gravity-really-caused-by-gravitational-time-dilation#comment137198_59097) on the previous question? $g_{00}$ is the purely time component of the spacetime metric matrix. (The convention is to index time as the zeroth dimension, not the fourth).  
> **username_2**: It's tricky getting an intuitive feel for this stuff. Our intuitions evolved to deal with speeds we normally encounter, eg the duration of a heartbeat feels comparable to the length of a pace, so 1 second ~ 1 metre. But in relativity, the natural speed is *c*, and it's hard to appreciate that 1 millisecond is the same magnitude as 300 km.  

---

## 1 answers

### Answer by username_3 on 2024-12-25. Score: 1 | Accepted answer

The other question was closed, so I put this here. One should watch these two PBS Space Time episodes that explain that mass causes time dilation, which is a curvature of space time, and with just the altered $g\_{tt}$ metric one has gravitational effects. Time dilation and reduced $g\_{tt}$ metric are nearly synonymous. In the second episode, he discusses that if light is a wave then gravity bends its path by time dilation, just as refraction bends light by changing its speed. But, if light is a particle one needs to see it as a curvature of space and time.

[Does time cause gravity](https://www.pbs.org/video/does-time-cause-gravity-00kr0q/)

[How does gravity affect light](https://www.pbs.org/video/how-does-gravity-affect-light-zx47ji/)

The motion of a particle in a weak static field can be found by starting with the geodesic equation of motion, a general solution of Einstein's field equations:

$$
\frac{\mathrm{d}^2 x^\alpha}{\mathrm{d}\tau^2} + \Gamma^\alpha\_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}\tau}\frac{\mathrm{d}x^\nu}{\mathrm{d}\tau} = 0
$$
where $x^\alpha$
is the 4-vector describing the position of a test particle and $\tau$ is proper time.

When the motion is far below from relativistic speeds, we can neglect $\frac{\mathrm{d}x^i}{\mathrm{d}\tau} (i=1..3$) compared to the $i=0$ component ($t=x^0$):

$$
\frac{\mathrm{d}^2 x^\alpha}{\mathrm{d}\tau^2} + \Gamma^\alpha\_{00}\big(\frac{\mathrm{d}t}{\mathrm{d}\tau}\big)^2=0.
$$

In a stationary gravitational field, all time derivatives vanish, thus only one of the three terms of the connection coefficients survives:

$$
\Gamma^\alpha\_{00} = -\frac{1}{2} g^{\mu\nu} \frac{\partial{g\_{00}}}{\partial{x^v}}.
$$

For weak gravitational fields the metric can be written as small perturbations, $h$, to the Minkowski metrics, $\eta$,

$$
g\_{\mu\nu} = \eta\_{\mu\nu} + h\_{\mu\nu}.
$$

Ignoring terms of second order in the $h\_{\mu\nu}$, we have
$$
\Gamma^\alpha\_{00} = -\frac{1}{2} \eta^{\mu\nu} \frac{\partial{h\_{00}}}{\partial{x^\nu}},
$$
which is just the gradient of the perturbation to time.

From this it is evident that the orbit in the weak field depends only on the perturbation to time, specifically $h\_{00}$ and $\frac{\mathrm{d}t}{\mathrm{d}\tau}$, the time dilation. And it is this which implies that such orbits depend only on curvatures in time dimension of 4 space.

Continuing on, if one sets $h\_{00} = 2 \phi$, one finds the geodesic equation is

$$
\frac{\mathrm{d}^2 \textbf{x}}{\mathrm{d}t^2} = - \nabla \phi
$$

which is Newtonian law. To express this as plainly as possible:
$$
\ddot{\textbf{x}} = - \frac{1}{2}\nabla h\_{00}
$$
which says acceleration is equal to minus one-half the gradient of the perturbation of the time metric.

Much of this is presented in the chapter "Newtonian Gravity in the Language of Curved Spacetime" in "Gravitation" by Misnor, Thorne, and Wheeler.

> **Comments:**
> **username_0**: Mite, I have seen 'em all, on Youtube, and I am even more confused, "MK"? I thought I understood GR and gravity, until this claim that gravity is really caused by GTD, started appearing, all over the place. I can see how GTD can deflect photons but how about matter/objects? Wouldn't GTD just make my feet a bit younger, than my head? If GTD is a measure of time curvature, then what is a simple measure of space curvature?  
> **username_3**: You are traveling through 4-space.  If your feet are traveling more slowly then you rotate in 4-space.  Your geodesic now points toward the gravity source.        We don’t have a distinct word for space curvature.  For non-relativistic motion c$\Delta t >> \Delta x$ so effects of space curvature are insignificant.   Curvature appears flat locally.  Precession of Mercury is affected though.  

---

