Understanding E subscript R Notation in the Context of General Relativity

In summary, "Understanding E subscript R Notation in the Context of General Relativity" explores the significance and application of the E subscript R notation, which denotes the energy density in the framework of general relativity. The notation is utilized to describe the distribution of energy and momentum in spacetime, linking it to Einstein's field equations. The article emphasizes how this notation aids in modeling gravitational effects and understanding the dynamics of various astrophysical phenomena, providing insights into the complex interplay between matter and the curvature of spacetime.
  • #1
robotkid786
22
7
Homework Statement
It's not homework, it's a query about the book "spacetime and geometry"
Relevant Equations
F=GMm/r^2 e subscript r in brackets
All I know is that e subscript r must be a vector cos the book says so, but what does it mean, is it, a konstant in vector form? I'm confused by it (page one, chapter one spacetime and geometry by SeanCaroll)

Help is appreciated

Edit. Is vector r describing the curvature that takes place ?
 
Physics news on Phys.org
  • #2
It's the unit vector in the radial direction, also written ##\hat r##.
 
  • Like
Likes robotkid786
  • #3
robotkid786 said:
Homework Statement: It's not homework, it's a query about the book "spacetime and geometry"
Relevant Equations: F=GMm/r^2 e subscript r in brackets

All I know is that e subscript r must be a vector cos the book says so, but what does it mean, is it, a konstant in vector form? I'm confused by it (page one, chapter one spacetime and geometry by SeanCaroll)

Help is appreciated

Edit. Is vector r describing the curvature that takes place ?
That's Carroll's notation for a unit vector in the direction of separation of the masses. The more usual notation involves the position vectors of the two masses ##\vec r_1, \vec r_2##. In which case, the force on mass ##m_2## is:
$$\vec F_2 = \frac{Gm_1m_2}{|\vec r_1 - \vec r_2|^3}(\vec r_1 - \vec r_2)$$
 
  • #4
haruspex said:
It's the unit vector in the radial direction, also written ##\hat r##.
That's not actually what Carroll means in this context.
 
  • Like
Likes Delta2
  • #5
haruspex said:
It's the unit vector in the radial direction, also written ##\hat r##.
If by radial direction you mean the direction of separation of masses M and m then ok, but if you mean the unit vector in polar, spherical or cylindrical coordinates, then you are not ok :D.
 
  • #6
Delta2 said:
If by radial direction you mean the direction of separation of masses M and m then ok, but if you mean the unit vector in polar, spherical or cylindrical coordinates, then you are not ok :D.
##e_r, e_\theta## are often used to denote unit vectors in the radial and tangential directions in polar coordinates. Likewise ##e_x## etc. in Cartesian. See e.g. https://www.chegg.com/homework-help...-z-r-right-rangle-unit-radial-vect-q114756124.

Per @PeroK, it seems that Sean Carroll is here, in effect, electing to take one of the masses as being at the origin.
 
  • #7
haruspex said:
Per @PeroK, it seems that Sean Carroll is here, in effect, electing to take one of the masses as being at the origin.
It's actually his own slighty idiosynchratic notation. In any case, it's only the prelude and only the briefest summary of Newtonian gravity before he introduces the Einstein Field Equations.
 
  • #8
So, does this mean. The hat notation is equivalent to the subscription notation here?

In which case, as above, the vector here being referred to is in reference to the circular distance between m and M?

Sorry if I'm getting it wrong. I havent even done vectors in my degree yet and I can hardly remember the topic either 🙃
 
  • #9
robotkid786 said:
So, does this mean. The hat notation is equivalent to the subscription notation here?

In which case, as above, the vector here being referred to is in reference to the circular distance between m and M?

Sorry if I'm getting it wrong. I havent even done vectors in my degree yet and I can hardly remember the topic either 🙃
If you haven't studied vectors yet, you are wasting your time with a graduate textbook on GR. Moreover, you cannot seriously learn GR until you have mastered SR.
 
  • #10
The vector being referred is the unit vector in the direction of the line that connects the centers of the two masses.

If we consider a coordinate system in spherical or polar coordinates and we put its origin in mass M or the mass m, then the vector being referred is also the radial unit vector of the coordinate system.
 
  • Like
Likes MatinSAR, robotkid786 and PeroK
  • #11
Got you, thanks man
 
  • Like
Likes Delta2
  • #12
PeroK said:
If you haven't studied vectors yet, you are wasting your time with a graduate textbook on GR. Moreover, you cannot seriously learn GR until you have mastered SR.
Don't cut the wings of possibly I have to admit overambitious young students.
 

FAQ: Understanding E subscript R Notation in the Context of General Relativity

What does E subscript R notation represent in the context of General Relativity?

In the context of General Relativity, E subscript R (ER) typically refers to the energy density associated with the Ricci curvature of spacetime. The Ricci curvature is a mathematical object that describes the degree to which the geometry of spacetime is curved by the presence of mass and energy.

How is E subscript R related to the Einstein Field Equations?

ER is related to the Einstein Field Equations (EFE), which are the fundamental equations in General Relativity describing how matter and energy influence spacetime curvature. Specifically, the EFE can be written as Gμν = 8πTμν, where Gμν is the Einstein tensor (which includes Ricci curvature) and Tμν is the stress-energy tensor. ER can be seen as a component of the energy-momentum distribution described by Tμν.

How do you calculate E subscript R in a given spacetime?

To calculate ER in a given spacetime, you need to solve the Einstein Field Equations for the specific metric describing that spacetime. Once the metric is known, you can compute the Ricci tensor (Rμν) and then determine the energy density ER from the components of the stress-energy tensor Tμν using the relationship established by the EFE.

What is the significance of E subscript R in cosmological models?

In cosmological models, ER is significant because it quantifies the energy density due to the curvature of spacetime, which affects the dynamics of the Universe's expansion. For instance, in the Friedmann-Lemaître-Robertson-Walker (FLRW) metric used in cosmology, the Ricci curvature plays a key role in determining the rate of expansion and the overall geometry of the Universe.

Can E subscript R be negative, and what would that imply physically?

Yes, ER can be negative, which would imply that the spacetime has a curvature associated with a repulsive gravitational effect. This can occur in scenarios involving exotic matter or energy, such as in the context of dark energy or certain solutions to the Einstein Field Equations that permit negative energy densities, potentially leading to phenomena like wormholes or the accelerated expansion of the Universe.

Similar threads

I Index notation of vector rotation
Replies
7
Views
228
Complex operations, bra-ket notation, confusion
Replies
1
Views
1K
I Principal and Gaussian curvature of the FRW metric
Replies
8
Views
184
I Confusion about index notation and operations of GR
Replies
8
Views
3K
I Understanding Ewald's sphere in the context of X-Ray diffraction
Replies
3
Views
4K
I Tangent space basis vectors under a coordinate change
Replies
12
Views
4K
I Passive Transformation and Rotation Matrix
Replies
1
Views
872
Understanding work in translation and rotation of a magnetic dipole
Replies
1
Views
1K
Kleppner - Example 1.18: Representation of Position Vector
Replies
9
Views
2K
Vector Mechanics Homework: Motion, Velocity & Acceleration
Replies
7
Views
4K

Hot Threads

Recent Insights

Back
Top