Arithmetic combinatorics: Difference between revisions

Revision as of 19:55, 29 December 2013

Newton–Cartan theory is a geometrical re-formulation, as well as a generalization, of Newtonian gravity developed by Élie Cartan. In this re-formulation, the structural similarities between Newton's theory and Albert Einstein's general theory of relativity are readily seen, and it has been used by Cartan and Kurt Friedrichs to give a rigorous formulation of the way in which Newtonian gravity can be seen as a specific limit of general relativity, and by Jürgen Ehlers to extend this correspondence to specific solutions of general relativity.

Geometric formulation of Poisson's equation

In Newton's theory of gravitation, Poisson's equation reads

\Delta U=4\pi G\rho \,

where $U$ is the gravitational potential, $G$ is the gravitational constant and $\rho$ is the mass density. The weak equivalence principle motivates a geometric version of the equation of motion for a point particle in the potential $U$

m_{t}{\ddot {\vec {x}}}=-m_{g}\nabla U

where $m_{t}$ is the inertial mass and $m_{g}$ the gravitational mass. Since, according to the weak equivalence principle $m_{t}=m_{g}$ , the according equation of motion

{\ddot {\vec {x}}}=-\nabla U

doesn't contain anymore a reference to the mass of the particle. Following the idea that the solution of the equation then is a property of the curvature of space, a connection is constructed so that the geodesic equation

{\frac {d^{2}x^{\lambda }}{ds^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {dx^{\mu }}{ds}}{\frac {dx^{\nu }}{ds}}=0

represents the equation of motion of a point particle in the potential $U$ . The resulting connection is

\Gamma _{\mu \nu }^{\lambda }=\gamma ^{\lambda \rho }U_{,\rho }\Psi _{\mu }\Psi _{\nu }

with $\Psi _{\mu }=\delta _{\mu }^{0}$ and $\gamma ^{\mu \nu }=\delta _{A}^{\mu }\delta _{B}^{\nu }\delta ^{AB}$ ( $A,B=1,2,3$ ). The connection has been constructed in one inertial system but can be shown to be valid in any inertial system by showing the invariance of $\Psi _{\mu }$ and $\gamma ^{\mu \nu }$ under Galilei-transformations. The Riemann curvature tensor in inertial system coordinates of this connection is then given by

R_{\kappa \mu \nu }^{\lambda }=2\gamma ^{\lambda \sigma }U_{,\sigma [\mu }\Psi _{\nu ]}\Psi _{\kappa }

where the brackets $A_{[\mu \nu ]}={\frac {1}{2!}}[A_{\mu \nu }-A_{\nu \mu }]$ mean the antisymmetric combination of the tensor $A_{\mu \nu }$ . The Ricci tensor is given by

R_{\kappa \nu }=\Delta U\Psi _{\kappa \nu }\,

which leads to following geometric formulation of Poisson's equation

R_{\mu \nu }=4\pi G\rho \Psi _{\mu }\Psi _{\nu }\,

Bargmann lift

It was shown that four-dimensional Newton–Cartan theory of gravitation can be reformulated as Kaluza–Klein reduction of five-dimensional Einstein gravity along a null-like direction.^[1] This lifting is considered to be useful for non-relativistic holographic models.^[2]

References

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Bibliography

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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

Chapter 1 of Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

↑ C. Duval, G. Burdet, H. P. Künzle, and M. Perrin, Bargmann structures and Newton-Cartan theory, Phys. Rev. D 31, 1841–1853 (1985)
↑ Walter D. Goldberger, AdS/CFT duality for non-relativistic field theory, JHEP03(2009)069 [1]

[1] C. Duval, G. Burdet, H. P. Künzle, and M. Perrin, Bargmann structures and Newton-Cartan theory, Phys. Rev. D 31, 1841–1853 (1985)

[2] Walter D. Goldberger, AdS/CFT duality for non-relativistic field theory, JHEP03(2009)069 [1]

[1]

[2]

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+'''Newton–Cartan theory''' is a geometrical re-formulation, as well as a generalization, of [[Newtonian gravity]] developed by [[Élie Cartan]].  In this re-formulation, the structural similarities between Newton's theory and [[Albert Einstein]]'s [[general theory of relativity]] are readily seen, and it has been used by Cartan and [[Kurt Friedrichs]] to give a rigorous formulation of the way in which Newtonian gravity can be seen as a specific limit of general relativity, and by [[Jürgen Ehlers]] to extend this correspondence to specific [[solutions]] of general relativity.
+==Geometric formulation of Poisson's equation==
+In Newton's theory of gravitation, [[Poisson's equation]] reads
+:<math>
+\Delta U = 4 \pi G \rho \,
+</math>
+where <math>U</math> is the gravitational potential, <math>G</math> is the gravitational constant and <math>\rho</math> is the mass density. The weak [[equivalence principle]] motivates a geometric version of the equation of motion for a point particle in the potential <math> U </math>
+:<math>
+m_t \ddot{\vec x} = - m_g \nabla U
+</math>
+where <math>m_t</math> is the inertial mass and <math>m_g</math> the gravitational mass. Since, according to the weak equivalence principle <math> m_t = m_g </math>, the according equation of motion
+:<math>
+\ddot{\vec x} = - \nabla U
+</math>
+doesn't contain anymore a reference to the mass of the particle. Following the idea that the solution of the equation then is a property of the curvature of space, a connection is constructed so that the [[geodesic equation]]
+:<math>
+\frac{d^2 x^\lambda}{d s^2} + \Gamma_{\mu \nu}^\lambda \frac{d x^\mu}{d s}\frac{d x^\nu}{d s} = 0
+</math>
+represents the equation of motion of a point particle in the potential <math>U</math>. The resulting connection is
+:<math>
+\Gamma_{\mu \nu}^{\lambda} = \gamma^{\lambda \rho} U_{, \rho} \Psi_\mu \Psi_\nu
+</math>
+with <math>\Psi_\mu = \delta_\mu^0 </math> and <math>\gamma^{\mu \nu} = \delta^\mu_A \delta^\nu_B \delta^{AB}</math> (<math> A, B = 1,2,3 </math>). The connection has been constructed in one inertial system but can be shown to be valid in any inertial system by showing the invariance of <math> \Psi_\mu</math> and <math> \gamma^{\mu \nu} </math> under Galilei-transformations. The Riemann curvature tensor in inertial system coordinates of this connection is then given by
+:<math>
+R^\lambda_{\kappa \mu \nu} = 2 \gamma^{\lambda \sigma} U_{, \sigma [ \mu}\Psi_{\nu]}\Psi_\kappa
+</math>
+where the brackets <math> A_{[\mu \nu]} = \frac{1}{2!} [ A_{\mu \nu} - A_{\nu \mu} ] </math> mean the antisymmetric combination of the tensor <math> A_{\mu \nu} </math>. The [[Ricci tensor]] is given by
+:<math>
+R_{\kappa \nu} = \Delta U \Psi_{\kappa \nu} \,
+</math>
+which leads to following geometric formulation of Poisson's equation
+:<math>
+R_{\mu \nu} = 4 \pi G \rho \Psi_\mu \Psi_\nu \,
+</math>
+==Bargmann lift==
+It was shown that four-dimensional Newton–Cartan theory of gravitation can be reformulated as [[Kaluza–Klein reduction]] of five-dimensional Einstein gravity along a null-like direction.<ref>C. Duval, G. Burdet, H. P. Künzle, and M. Perrin, Bargmann structures and Newton-Cartan theory, Phys. Rev. D 31, 1841–1853 (1985)</ref> This lifting is considered to be useful for non-relativistic [[Holographic principle|holographic]] models.<ref>Walter D. Goldberger, AdS/CFT duality for non-relativistic field theory, JHEP03(2009)069 [http://lanl.arxiv.org/abs/0806.2867]</ref>
+==References==
+{{Reflist}}
+==Bibliography==
+*{{Citation
+ | last=Cartan
+ | first=Elie
+ | year=1923
+ | journal=Ann. Ecole Norm.
+ | volume=40
+ | page=325
+ }}
+*{{Citation
+ | last=Cartan
+ | first=Elie
+ | year=1924
+ | journal=Ann. Ecole Norm.
+ | volume=41
+ | page=1
+ }}
+*{{Citation
+ | last=Cartan
+ | first=Elie
+ | year=1955
+ | title=OEuvres Complétes
+ | volume=III/1
+ | pages=659, 799
+ | publisher=Gauthier-Villars
+ }}
+*Chapter 1 of {{Citation
+ | last=Ehlers
+ | first = Jürgen
+ | author-link=Jürgen Ehlers
+ | contribution=Survey of general relativity theory
+ | editor-last=Israel
+ | editor-first=Werner
+ | title=Relativity, Astrophysics and Cosmology
+ | year=1973
+ | publisher=D. Reidel
+ | pages=1–125
+ | isbn=90-277-0369-8
+ }}
+{{DEFAULTSORT:Newton-Cartan theory}}
+[[Category:Theories of gravitation]]

Arithmetic combinatorics: Difference between revisions

Revision as of 19:55, 29 December 2013

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Geometric formulation of Poisson's equation

Bargmann lift

References

Bibliography

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Revision as of 19:55, 29 December 2013

Geometric formulation of Poisson's equation

Bargmann lift

References

Bibliography

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