- #1

Grasshopper

Gold Member

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- TL;DR Summary
- I want to knock this question out once and for all.

I've heard it and I've read* it before, so I just want to make sure I understand this so I never have to wonder about it again.

So, are tidal forces exactly curvature of space?

Here's why I think the answer to that is yes:

.I've seen a spacetime interval equation which has a coefficient on the time term, and a coefficient on the space terms. Further, I'm told* that if you ignore the coefficient on the space components (or rather, set them to 1), it gives an equation that produces the results of Newton's model. Specifically, the equation (with the spatial coefficient set to 1) is this (unless I'm mistaken):

##Δs^2 = (1 - \frac{2GM}{c^2r})(cΔt)^2 - (Δx)^2 - (Δy)^2 - (Δz)^2##

and I have read that the coefficient here shows how time would be curved.

I interpret the equation as this: As r decreases, the entire time term grows (the bigger r gets, the smaller ##\frac{2GM}{c^2r}## gets, so the larger the coefficient is). Intuitively, that means to me that the "tick marks" on time are not evenly spaced as a function of r. They "crunch" as if being "curved."

However, I am told when the coefficient on the space terms is not 1, space curves as well:

##Δs^2 = (1 - \frac{2GM}{c^2r})(cΔt)^2 - (1 + \frac{2GM}{c^2r})[(Δx)^2 + (Δy)^2 + (Δz)^2]##

However, the spatial term is missing a factor of ##c^2##, which I would interpret as a reason that tidal forces are small (at least here on Earth).Now, please feel free to eviscerate all this intuitive interpretation if it's way off. I would truly appreciate that, because the goal here is to learn. But whatever parts are close to right, please let me know as well.But most importantly, are tidal forces curved space?One final thought: Another interpretation I'm thinking about is that the entire equation relates to tidal forces, both the space and time components. That also makes sense to me because if the equation with the space coefficients set to 1 approximates Newton, then the time component must also include tidal forces, because presumably Newton includes tidal forces. So in that case, spacetime curvature is tidal forces, not just space curvature.Thanks as always!.

.

*"Read" in particular refers to the source below, and "I am told" here means things I've seen on various physics forums and the rest of the internet, the place where you definitely shouldn't be trying to learn physics, which is why I'm here trying to see if it's right. The only forum I trust is this one. One particular source is from Brown University:

https://www.math.brown.edu/tbanchof/STG/ma8/papers/dstanke/Project/Relativity.html

"When he returned to the problem, he focused on a gap in his previous reasoning: he had ignored

(bold is my emphasis)

Given my last adventure with posting something from what ostensibly is a reputable source [a QM article from a Journal of Physics: Conference Series], I am not fully trusting it until the experts here confirm or reject it.

So, are tidal forces exactly curvature of space?

Here's why I think the answer to that is yes:

.I've seen a spacetime interval equation which has a coefficient on the time term, and a coefficient on the space terms. Further, I'm told* that if you ignore the coefficient on the space components (or rather, set them to 1), it gives an equation that produces the results of Newton's model. Specifically, the equation (with the spatial coefficient set to 1) is this (unless I'm mistaken):

##Δs^2 = (1 - \frac{2GM}{c^2r})(cΔt)^2 - (Δx)^2 - (Δy)^2 - (Δz)^2##

and I have read that the coefficient here shows how time would be curved.

I interpret the equation as this: As r decreases, the entire time term grows (the bigger r gets, the smaller ##\frac{2GM}{c^2r}## gets, so the larger the coefficient is). Intuitively, that means to me that the "tick marks" on time are not evenly spaced as a function of r. They "crunch" as if being "curved."

However, I am told when the coefficient on the space terms is not 1, space curves as well:

##Δs^2 = (1 - \frac{2GM}{c^2r})(cΔt)^2 - (1 + \frac{2GM}{c^2r})[(Δx)^2 + (Δy)^2 + (Δz)^2]##

However, the spatial term is missing a factor of ##c^2##, which I would interpret as a reason that tidal forces are small (at least here on Earth).Now, please feel free to eviscerate all this intuitive interpretation if it's way off. I would truly appreciate that, because the goal here is to learn. But whatever parts are close to right, please let me know as well.But most importantly, are tidal forces curved space?One final thought: Another interpretation I'm thinking about is that the entire equation relates to tidal forces, both the space and time components. That also makes sense to me because if the equation with the space coefficients set to 1 approximates Newton, then the time component must also include tidal forces, because presumably Newton includes tidal forces. So in that case, spacetime curvature is tidal forces, not just space curvature.Thanks as always!.

.

*"Read" in particular refers to the source below, and "I am told" here means things I've seen on various physics forums and the rest of the internet, the place where you definitely shouldn't be trying to learn physics, which is why I'm here trying to see if it's right. The only forum I trust is this one. One particular source is from Brown University:

https://www.math.brown.edu/tbanchof/STG/ma8/papers/dstanke/Project/Relativity.html

"When he returned to the problem, he focused on a gap in his previous reasoning: he had ignored

**tidal forces**. Tidal forces are the name given to the forces that result from differences in the strength of the gravitational forces on an object. He had ignored these forces in his earlier work, but their existence invalidated his theory because they allow someone in free-fall to observe the effects of gravity; if you observe your body to be stretching to enormous length, you can be sure you're being pulled by gravity and not floating in empty space. How could Einstein explain tidal forces?**The answer lies in****curved space**."(bold is my emphasis)

Given my last adventure with posting something from what ostensibly is a reputable source [a QM article from a Journal of Physics: Conference Series], I am not fully trusting it until the experts here confirm or reject it.

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