Subject abstract:
In this work, I will attempt to give a brief history on time and why today physics appears to be indicating that time doesn't even exist. We will soon come to understand in my work, that we are basing this on the very foundation of relativity in which ultimately all events happen side by side and that ''time'' as we know it is nothing but, what Einstein called an ''illusion.'' We will find out that this ''illusion'' is carried on to the very beginning of the universe itself; if it had a beginning that is.
The Problem of Time
The so-called ''problem of time'' as Julian Barbour and Fotini Markopoulou calls it [1] has been one been around for a while in physics. It involves a curious observation about the quantized form of Einsteins field equations. The equation which described this ''quantum gravity equation,'' was formulated by Bryce de Witt for Wheeler who worked with him on the theory.
It turned out to be the most controversial equation ever made because it seemed to say that time on the whole did not exist. Let's take a look at this equation
$$H | \Psi > = 0$$
Let's get familiar with the notation. $$\Psi$$ is the wave function of the universe. It is manifestly global. $$H$$ is the total energy content of the universe. So what's the big deal?
The holistic dynamics of the universe appear to be time independant: The equation normally would look like a time-dependant Schrodinger equation but we don't have this in this case, also it isn't complex which is an interesting observation, for what does it mean that quantum gravity is not inherently a complex field? There are of course interesting relationships between equations which have time and are inherently complex but this is too deep a subject for this moment so it will be discussed below in references [*].
What does it mean that General Relativity predicts there being no time? First thing people get wrong, is that they think if time doesn't exist, then change cannot happen. Oh contraire! Things can change, it's time that's the superfluous additive. Time has to do with things changing in space, it's not space itself, which is one of the reasons why Minkowski's unification of the two will eventually turn out to be inconsistent.
It may even turn out that space is also not fundamental; in fact there have been some scientists who have forwarded this question over the years. It would also mean that separation of objects are an illusion, which I'd probably tend to agree with due to non-locality of quantum entanglement. If separation of objects where an illusion, it would go hand in hand with the proposal that there is no such thing as a separation of events in relativity which we somewhat find ourselves forced to think of. In relativity, the past, present and future are all illusions and if it wasn't for this illusion we encounter, all events would happen simultaneously.
This is the reality of the problem of time in relativity. Not everyone is aware of it but it has persisted since the early pioneers who first quantized Einstein's equations in 1967 and actually remains... a somewhat quiet problem today.
[1] - http://www.youtube.com/watch?v=I5rExaKLEoU
F. Markopoulou 'Space does not exist, so time can.' http://fqxi.org/community/forum/topic/376
Time as the Measure of Change
Time isn't exactly disappearing, we are just unveiling it for it's true form and that is that it measures changes in the universe. The universe as a closed system to define time, we need moving clocks (matter). Clocks where formed originally to measure what we call time, but time is a concept created to understand the linear form of events which come about through the physical changes of the system. There is in fact no indication time even exists external to the human perception. We already have wonderful biological reasons as to why we even have a sense of time and we owe this to a set of gene regulators inside the brain which regulates our ''flow'' of time.
If there was no motion in the universe, there would be no way to even say time passed. Time in this sense is created to measure an order of change in the universe. Einstein certainly, when developing the relativity theories never considered time as anything more special than a phenomenon which a clock measures. Motion in General relativity arises as a symmetry of the theory, it isn't a true time evolution [1]. The Einstein field equations are invariant under diffeomorphisms in which we actually find that spacetime points are not physical in themselves, only events are physical characterized by physical interactions.
[1] http://fqxi.org/data/essay-contest-...f?phpMyAdmin=0c371ccdae9b5ff3071bae814fb4f9e9
Geometrogenesis
We can find some interesting ways to view this picture we have elaborated on concerning the illusion of time. The model extends to geometrogensis, a theory of how geometry and matter comes into the universe. Geometry and matter are relatively ''late phenomenon'' and are classed as low energy phenomenon. As you get closer to the big bang, geometry begins to vanish completely along with matter and you no longer have time as we deal with it in physics. Time is therefore a low energy phenomenon experienced by moving clocks.
It also means as you would expect that time is not fundamental, it is an emergent property of a low energy epoch. We may also expect a serious theory which extends this idea would treat gravity also as an emergent property.
Removing Time
In Julian Barbours attempt to convince the world of these idea's, he showed that removing time from equations is no problem at all. In fact, in his ''fundamental'' equation, you remove time by making a square root! Very simply, the equation given is
$$\delta t: = \frac{sqrt{\sum m_i \delta x_i \cdot \delta x_i}}{\sqrt{2(E-U)}}$$
On the right hand side Barbour argues [1] that the right hand side describes a change which becomes the definition of time. It's an interesting kinematical equation because it purports to recover Newtonian equations as well. Each particle ‘advances time’ in proportion to the square root of its mass and to its displacement, the total contribution $$\delta s$$ which is weighed by $$\sqrt{2(E-U)}$$.
Let's derive something in style of Barbour, a description of a system which changes without the specification of any time variable.
The plane wave solution to the Schrodinger equation is
$$\Psi = e^{i(k \cdot r - \omega t)}$$
The gradient of this is
$$\hat{e}_x \frac{\partial \psi}{\partial x} + \hat{e}_y \frac{\partial \psi}{\partial y} + \hat{e}_z \frac{\partial \psi}{\partial z} = ik_x \psi e_x + ik_y \psi e_y + ik_z \psi e_z$$
$$= \frac{i}{\hbar}(p_x e_x + p_y e_y + p_z e_z)\psi = \frac{i}{\hbar} \hat{P}\psi$$
and because $$e_x, e_y$$ and $$e_z$$ are the base space, we can see that
$$\hat{P} = -i \hbar \nabla$$
Which makes our momentum in position space.
Keeping in mind that
$$\frac{\partial}{\partial x}(\frac{\partial}{\partial x}) = \frac{\partial^2}{\partial x^2}$$
Then
$$-i \hbar \nabla(\frac{\partial}{\partial y}+ \frac{\partial}{\partial x} + \frac{\partial}{\partial z})$$
Keeping this all in mind, we will now move on to the rest of the derivation.
Consider a simple spacetime interval as:
$$d\tau^2 = dt^2 - d\vec{x}^2$$
Where we have set $$c=1$$ in this case. You actually calculate the length of a worldline by taking into consideration the integral
$$L(W) = \int_W d\tau$$
You can, it was shown to me a while ago now, that worldines can be written in terms of time by the chain rule. Doing so, you can rewrite the time derivatives as dots on your variables and can end up with
$$L(W) = \int_{t_0}^{t_1} \sqrt{1 - ||\dot{x}||^2}\ dt$$
From here, you would calculate the Langrangian by simply multiplying mass into the equation, so we would have
$$\mathcal{L} = -M\sqrt{1 - ||\dot{x}||^2}$$
Now using the generalized velocity,
$$\mathcal{L}(\dot{q}\dot{q}) = -M\sqrt{1 - \dot{q}\dot{q}}$$
The canonical momentum can be written as
$$\frac{\partial \mathcal{L}}{\partial \dot{q}} = \frac{M\dot{q}}{\sqrt{1 - \dot{q}\dot{q}}}$$
This is relativistic and is incorporated as one can see, into the idea of worldlines. Now, in my equation, I decided to multiply the momentum with distance. Of course, this was just the quantum action $$\hbar$$, but ignoring that fact for now, we wish to calculate the distance really as a displacement of all the particles in the universe $$d_i$$ using Barbour's approach. Doing so, will require an integral.
Taking the integral of the equation ''slices'' a worldline the distance will be small $$\delta d$$ as a displacement measuring a change in the system. Peicing together our terms
$$\frac{\partial \mathcal{L}}{\partial \dot{q}}$$
$$\nabla^2 \psi$$
We can make our desired expression
$$\frac{\partial \mathcal{L}}{\partial \dot{q}}(\delta d_i) \nabla^2 \psi$$
While this expression is essentially about dynamics which describes a system changing, to be accurate, it describes a mass flow $$(MT^{-1})$$.
[1] - http://arxiv.org/pdf/gr-qc/0309089v1.pdf
Quantum Cosmological Applications
Let's begin to understand what this theory means... if it is true then this has some very important applications to the universe and to how we understand it. If time doesn't exist, then can we say there is a beginning to the universe?
In this work and also in Barbours work, we argue that there is only change in the universe there is no such thing as time. In this case we might be able to argue there was an initiaI condition in which all change came from, but this structured ordered idea of how to think of the big bang becomes less favourable when you realize relativity forbids in it's over-all stucture a meaningful set of events in time. Instead, in relativity we are somewhat forced to think of events happening all at once. This means you don't have this order, which makes it difficult to talk about the big bang as a beginning. Because order breaks down without a past, present and future, the big bang can no more successfully describe a beginning than the big crunch [1]. In the reference just given, physicist Wolf describes relativity as making the big bang and big crunch as almost simultaneous events - if just a flutter between the two.
The problem just remains like a stubborn stain when you realize that quantizing General Relativity leads to the ''problem of time'' characterized by a Schrodinger-like equation but is absolutely devoid of any description of time.
In fact, Julian Barbour has made it clear this shouldn't surprise us. There are in fact many time independent equations in physics and in fact the evolution equation which governs wavelike behaviour (the Schrodinger Equation) was originally developed in it's time independent form.
The quantization of the equations lead to the Wheeler de Witt equation, which is a theory of quantum gravity; without time. And if you don't have time, how can we measure whether the universe conserves energy? Actually, if it turns out that the universe is timeless and is allowed to spill energy then there maybe ways to experimentally test this. In fact, I have been aware of over the years that there have been serious efforts to measure whether the universe is leaking energy [2].
So the problem of time runs deeper than just being able to order events. Also, we should keep in mind, that it isn't only relativity which predicts that the order of events as you near the big bang should become obscure. In Geometrogenesis, time only exists in relativity in the form Minkowski offered by making it a space dimension and therefore it exists as part of the geometry of the vacuum. As you wind the ''clock'' of the universe back to it's initial stages, we begin to find geometry disappearing, at least in the view we tend to think of it. In reality, there is a singularity present and an infinite amount of curvature but it will be taken in this work that infinities do not purport to physical systems. In the initial beginning of the universe, we view it as a dimensionless point so there cannot be any time present because there is no space to talk about either.
Again, if time begins to disappear, we can no longer talk about the order of events never mind, the beginning of the universe. Time also appears when matter appears in geometrogenesis and in relativity theory. Before matter appeared with geometry, the universe dealt with only radiation fields. Since time doesn't pass for these fields, it's also hard in this respect how to talk about time.
[1] - Parallel Universes (Wolf) 1985
[2] - http://www.scientificamerican.com/article/is-the-universe-leaking-energy/
full article http://www.physics.uq.edu.au/download/tamarad/papers/SciAm_Energy.pdf
by physicist, T. Davis
Conclusions
It seemed that Einstein never considered time an objective phenomenon. Nor did Dirac for that matter who when pressed about the question to whether he thought spacetime was a fundamental union, he said he was inclined not to believe so [1]. Though Minkowski no doubt believed his ''union'' of space and time was a beautiful object he no doubt equally believed that it was describing reality; what this work has done it to show that time probably isn't as fundamental as we might think.
[1] http://www.youtube.com/watch?v=kwqkdvKHTlg
[*] additional references
http://arxiv.org/abs/1206.6290
http://arxiv.org/abs/hep-th/0601234
Complexification of the quantum fields
In the ideal world, we would not only have a time derivative, we would also have complexification acting as a coefficient on $$(\frac{\partial}{\partial t})$$. When we talk about complexifying fields, it can mean quite a few things. Let's begin with an example which can be given concerning our formulation.
We would have a Hamiltonian which contains a constraint:
$$\pi_t + H = 0$$
$$\pi_t$$ is the momentum conjugate to time (it's not a true conjugate) [1], it is like it. To get a quantum theory from this classical constraint into a wave equation, $$\pi_t$$ would become $$i \frac{\partial}{\partial t}$$. From there you would end up with the Schrodinger equation. From here, Barbour has made a very important point in our usual understanding of quantum mechanics... that being ''quantum mechanics is inherently complex.''
Quick lecture on Einsteins Field Equations
The thing which calculates curvature in General Relativity is the Riemann Tensor and its given as
$$R^{\rho}_{\sigma \mu \nu} = \partial_{\mu} \Gamma^{\rho}_{\mu \sigma} - \partial^{\rho}_{\nu \sigma} + \Gamma^{\rho}_{\mu \lambda} \Gamma^{\lambda}_{\nu \sigma} - \Gamma^{\rho}_{\nu \lambda} \Gamma^{\lambda}_{\mu \sigma}$$
The part $$\Gamma_{\mu}\Gamma_{\nu}$$ is what you call the commutator of two matrices.
You can rewrite it more compactly when you bracket expressions and realize that these are the derivatives of the connection ''Gamma''
$$\frac{\partial \Gamma_{\mu}}{\partial x^{\nu}} - \frac{\partial \Gamma_{\nu}}{\partial x^{\mu}} + \Gamma_{\nu}\Gamma_{\mu} - \Gamma_{\mu}\Gamma_{\nu}$$
You can only get the Riemann tensor by contracting the ''Ricci Tensor''. Notice that one alpha is on the upper indices and one is on the lower indices:
$$R_{\mu \nu} = R^{\alpha}_{\mu \alpha \nu}$$
Repeated indices means you automatically sum over these indices. The lowercase $$\mu \alpha$$ actually describe some rotation plain for a very small area displacement $$(dx^{\nu}, dx^{\mu})$$
You can also contract using the metric, for instance
$$R_{\lambda \sigma \mu \nu} = g_{\lambda \rho} R^{\rho}_{\sigma \mu \nu}$$
Can you guess which one is contracted? If you said $$\rho$$, you'd be right. What is $$g_{\mu \nu}$$ contracted with $$R^{\mu \nu}$$? It's just $$R$$ is the answer. You would get the curvature scalar by contracted the Ricci Tensor $$R^{\mu \nu}$$ and has this following form
$$\nabla_{\mu} R^{\mu \nu} = \frac{1}{2} g^{\mu \nu} \partial_{\mu} R$$
where we call $$\nabla_{\mu}$$ the covariant derivative. I think the covariant derivative originally came from work on fibre bundles. The property of a covariant derivative just has this form:
$$\nabla_{\mu}AB = A\nabla B + (\nabla A) B$$
The covariant derivative of $$g_{\mu \nu}$$ is actually zero.
$$\nabla_{\mu} R^{\mu \nu} = \frac{1}{2} \nabla_{\mu}g^{\mu \nu} R$$
$$\nabla [R^{\mu \nu} - \frac{1}{2}g^{\mu \nu} R ]= 0$$
This can be rewritten as a short-hand
$$R^{\mu \nu} - \frac{1}{2}g^{\mu \nu} R = G^{\mu \nu}$$
so
$$\nabla_{\mu}G^{\mu \nu} = 0$$
This is the local continuity equation for gravitational energy. As I said before, $$g^{\mu \nu}$$ derivative is zero, so what we have is
$$R - 2R = 0$$
and $$R=0$$ when there is no energy-momentum present.
So we learned the ''Einstein Tensor'' $$\nabla_{\mu} G^{\mu \nu}=0$$
The right hand side of $$\nabla_{\mu} R^{\mu \nu} = \frac{1}{2} \nabla_{\mu}g^{\mu \nu} R$$ describes the matter in a universe.
In this work, I will attempt to give a brief history on time and why today physics appears to be indicating that time doesn't even exist. We will soon come to understand in my work, that we are basing this on the very foundation of relativity in which ultimately all events happen side by side and that ''time'' as we know it is nothing but, what Einstein called an ''illusion.'' We will find out that this ''illusion'' is carried on to the very beginning of the universe itself; if it had a beginning that is.
The Problem of Time
The so-called ''problem of time'' as Julian Barbour and Fotini Markopoulou calls it [1] has been one been around for a while in physics. It involves a curious observation about the quantized form of Einsteins field equations. The equation which described this ''quantum gravity equation,'' was formulated by Bryce de Witt for Wheeler who worked with him on the theory.
It turned out to be the most controversial equation ever made because it seemed to say that time on the whole did not exist. Let's take a look at this equation
$$H | \Psi > = 0$$
Let's get familiar with the notation. $$\Psi$$ is the wave function of the universe. It is manifestly global. $$H$$ is the total energy content of the universe. So what's the big deal?
The holistic dynamics of the universe appear to be time independant: The equation normally would look like a time-dependant Schrodinger equation but we don't have this in this case, also it isn't complex which is an interesting observation, for what does it mean that quantum gravity is not inherently a complex field? There are of course interesting relationships between equations which have time and are inherently complex but this is too deep a subject for this moment so it will be discussed below in references [*].
What does it mean that General Relativity predicts there being no time? First thing people get wrong, is that they think if time doesn't exist, then change cannot happen. Oh contraire! Things can change, it's time that's the superfluous additive. Time has to do with things changing in space, it's not space itself, which is one of the reasons why Minkowski's unification of the two will eventually turn out to be inconsistent.
It may even turn out that space is also not fundamental; in fact there have been some scientists who have forwarded this question over the years. It would also mean that separation of objects are an illusion, which I'd probably tend to agree with due to non-locality of quantum entanglement. If separation of objects where an illusion, it would go hand in hand with the proposal that there is no such thing as a separation of events in relativity which we somewhat find ourselves forced to think of. In relativity, the past, present and future are all illusions and if it wasn't for this illusion we encounter, all events would happen simultaneously.
This is the reality of the problem of time in relativity. Not everyone is aware of it but it has persisted since the early pioneers who first quantized Einstein's equations in 1967 and actually remains... a somewhat quiet problem today.
[1] - http://www.youtube.com/watch?v=I5rExaKLEoU
F. Markopoulou 'Space does not exist, so time can.' http://fqxi.org/community/forum/topic/376
Time as the Measure of Change
Time isn't exactly disappearing, we are just unveiling it for it's true form and that is that it measures changes in the universe. The universe as a closed system to define time, we need moving clocks (matter). Clocks where formed originally to measure what we call time, but time is a concept created to understand the linear form of events which come about through the physical changes of the system. There is in fact no indication time even exists external to the human perception. We already have wonderful biological reasons as to why we even have a sense of time and we owe this to a set of gene regulators inside the brain which regulates our ''flow'' of time.
If there was no motion in the universe, there would be no way to even say time passed. Time in this sense is created to measure an order of change in the universe. Einstein certainly, when developing the relativity theories never considered time as anything more special than a phenomenon which a clock measures. Motion in General relativity arises as a symmetry of the theory, it isn't a true time evolution [1]. The Einstein field equations are invariant under diffeomorphisms in which we actually find that spacetime points are not physical in themselves, only events are physical characterized by physical interactions.
[1] http://fqxi.org/data/essay-contest-...f?phpMyAdmin=0c371ccdae9b5ff3071bae814fb4f9e9
Geometrogenesis
We can find some interesting ways to view this picture we have elaborated on concerning the illusion of time. The model extends to geometrogensis, a theory of how geometry and matter comes into the universe. Geometry and matter are relatively ''late phenomenon'' and are classed as low energy phenomenon. As you get closer to the big bang, geometry begins to vanish completely along with matter and you no longer have time as we deal with it in physics. Time is therefore a low energy phenomenon experienced by moving clocks.
It also means as you would expect that time is not fundamental, it is an emergent property of a low energy epoch. We may also expect a serious theory which extends this idea would treat gravity also as an emergent property.
Removing Time
In Julian Barbours attempt to convince the world of these idea's, he showed that removing time from equations is no problem at all. In fact, in his ''fundamental'' equation, you remove time by making a square root! Very simply, the equation given is
$$\delta t: = \frac{sqrt{\sum m_i \delta x_i \cdot \delta x_i}}{\sqrt{2(E-U)}}$$
On the right hand side Barbour argues [1] that the right hand side describes a change which becomes the definition of time. It's an interesting kinematical equation because it purports to recover Newtonian equations as well. Each particle ‘advances time’ in proportion to the square root of its mass and to its displacement, the total contribution $$\delta s$$ which is weighed by $$\sqrt{2(E-U)}$$.
Let's derive something in style of Barbour, a description of a system which changes without the specification of any time variable.
The plane wave solution to the Schrodinger equation is
$$\Psi = e^{i(k \cdot r - \omega t)}$$
The gradient of this is
$$\hat{e}_x \frac{\partial \psi}{\partial x} + \hat{e}_y \frac{\partial \psi}{\partial y} + \hat{e}_z \frac{\partial \psi}{\partial z} = ik_x \psi e_x + ik_y \psi e_y + ik_z \psi e_z$$
$$= \frac{i}{\hbar}(p_x e_x + p_y e_y + p_z e_z)\psi = \frac{i}{\hbar} \hat{P}\psi$$
and because $$e_x, e_y$$ and $$e_z$$ are the base space, we can see that
$$\hat{P} = -i \hbar \nabla$$
Which makes our momentum in position space.
Keeping in mind that
$$\frac{\partial}{\partial x}(\frac{\partial}{\partial x}) = \frac{\partial^2}{\partial x^2}$$
Then
$$-i \hbar \nabla(\frac{\partial}{\partial y}+ \frac{\partial}{\partial x} + \frac{\partial}{\partial z})$$
Keeping this all in mind, we will now move on to the rest of the derivation.
Consider a simple spacetime interval as:
$$d\tau^2 = dt^2 - d\vec{x}^2$$
Where we have set $$c=1$$ in this case. You actually calculate the length of a worldline by taking into consideration the integral
$$L(W) = \int_W d\tau$$
You can, it was shown to me a while ago now, that worldines can be written in terms of time by the chain rule. Doing so, you can rewrite the time derivatives as dots on your variables and can end up with
$$L(W) = \int_{t_0}^{t_1} \sqrt{1 - ||\dot{x}||^2}\ dt$$
From here, you would calculate the Langrangian by simply multiplying mass into the equation, so we would have
$$\mathcal{L} = -M\sqrt{1 - ||\dot{x}||^2}$$
Now using the generalized velocity,
$$\mathcal{L}(\dot{q}\dot{q}) = -M\sqrt{1 - \dot{q}\dot{q}}$$
The canonical momentum can be written as
$$\frac{\partial \mathcal{L}}{\partial \dot{q}} = \frac{M\dot{q}}{\sqrt{1 - \dot{q}\dot{q}}}$$
This is relativistic and is incorporated as one can see, into the idea of worldlines. Now, in my equation, I decided to multiply the momentum with distance. Of course, this was just the quantum action $$\hbar$$, but ignoring that fact for now, we wish to calculate the distance really as a displacement of all the particles in the universe $$d_i$$ using Barbour's approach. Doing so, will require an integral.
Taking the integral of the equation ''slices'' a worldline the distance will be small $$\delta d$$ as a displacement measuring a change in the system. Peicing together our terms
$$\frac{\partial \mathcal{L}}{\partial \dot{q}}$$
$$\nabla^2 \psi$$
We can make our desired expression
$$\frac{\partial \mathcal{L}}{\partial \dot{q}}(\delta d_i) \nabla^2 \psi$$
While this expression is essentially about dynamics which describes a system changing, to be accurate, it describes a mass flow $$(MT^{-1})$$.
[1] - http://arxiv.org/pdf/gr-qc/0309089v1.pdf
Quantum Cosmological Applications
Let's begin to understand what this theory means... if it is true then this has some very important applications to the universe and to how we understand it. If time doesn't exist, then can we say there is a beginning to the universe?
In this work and also in Barbours work, we argue that there is only change in the universe there is no such thing as time. In this case we might be able to argue there was an initiaI condition in which all change came from, but this structured ordered idea of how to think of the big bang becomes less favourable when you realize relativity forbids in it's over-all stucture a meaningful set of events in time. Instead, in relativity we are somewhat forced to think of events happening all at once. This means you don't have this order, which makes it difficult to talk about the big bang as a beginning. Because order breaks down without a past, present and future, the big bang can no more successfully describe a beginning than the big crunch [1]. In the reference just given, physicist Wolf describes relativity as making the big bang and big crunch as almost simultaneous events - if just a flutter between the two.
The problem just remains like a stubborn stain when you realize that quantizing General Relativity leads to the ''problem of time'' characterized by a Schrodinger-like equation but is absolutely devoid of any description of time.
In fact, Julian Barbour has made it clear this shouldn't surprise us. There are in fact many time independent equations in physics and in fact the evolution equation which governs wavelike behaviour (the Schrodinger Equation) was originally developed in it's time independent form.
The quantization of the equations lead to the Wheeler de Witt equation, which is a theory of quantum gravity; without time. And if you don't have time, how can we measure whether the universe conserves energy? Actually, if it turns out that the universe is timeless and is allowed to spill energy then there maybe ways to experimentally test this. In fact, I have been aware of over the years that there have been serious efforts to measure whether the universe is leaking energy [2].
So the problem of time runs deeper than just being able to order events. Also, we should keep in mind, that it isn't only relativity which predicts that the order of events as you near the big bang should become obscure. In Geometrogenesis, time only exists in relativity in the form Minkowski offered by making it a space dimension and therefore it exists as part of the geometry of the vacuum. As you wind the ''clock'' of the universe back to it's initial stages, we begin to find geometry disappearing, at least in the view we tend to think of it. In reality, there is a singularity present and an infinite amount of curvature but it will be taken in this work that infinities do not purport to physical systems. In the initial beginning of the universe, we view it as a dimensionless point so there cannot be any time present because there is no space to talk about either.
Again, if time begins to disappear, we can no longer talk about the order of events never mind, the beginning of the universe. Time also appears when matter appears in geometrogenesis and in relativity theory. Before matter appeared with geometry, the universe dealt with only radiation fields. Since time doesn't pass for these fields, it's also hard in this respect how to talk about time.
[1] - Parallel Universes (Wolf) 1985
[2] - http://www.scientificamerican.com/article/is-the-universe-leaking-energy/
full article http://www.physics.uq.edu.au/download/tamarad/papers/SciAm_Energy.pdf
by physicist, T. Davis
Conclusions
It seemed that Einstein never considered time an objective phenomenon. Nor did Dirac for that matter who when pressed about the question to whether he thought spacetime was a fundamental union, he said he was inclined not to believe so [1]. Though Minkowski no doubt believed his ''union'' of space and time was a beautiful object he no doubt equally believed that it was describing reality; what this work has done it to show that time probably isn't as fundamental as we might think.
[1] http://www.youtube.com/watch?v=kwqkdvKHTlg
[*] additional references
http://arxiv.org/abs/1206.6290
http://arxiv.org/abs/hep-th/0601234
Complexification of the quantum fields
In the ideal world, we would not only have a time derivative, we would also have complexification acting as a coefficient on $$(\frac{\partial}{\partial t})$$. When we talk about complexifying fields, it can mean quite a few things. Let's begin with an example which can be given concerning our formulation.
We would have a Hamiltonian which contains a constraint:
$$\pi_t + H = 0$$
$$\pi_t$$ is the momentum conjugate to time (it's not a true conjugate) [1], it is like it. To get a quantum theory from this classical constraint into a wave equation, $$\pi_t$$ would become $$i \frac{\partial}{\partial t}$$. From there you would end up with the Schrodinger equation. From here, Barbour has made a very important point in our usual understanding of quantum mechanics... that being ''quantum mechanics is inherently complex.''
Quick lecture on Einsteins Field Equations
The thing which calculates curvature in General Relativity is the Riemann Tensor and its given as
$$R^{\rho}_{\sigma \mu \nu} = \partial_{\mu} \Gamma^{\rho}_{\mu \sigma} - \partial^{\rho}_{\nu \sigma} + \Gamma^{\rho}_{\mu \lambda} \Gamma^{\lambda}_{\nu \sigma} - \Gamma^{\rho}_{\nu \lambda} \Gamma^{\lambda}_{\mu \sigma}$$
The part $$\Gamma_{\mu}\Gamma_{\nu}$$ is what you call the commutator of two matrices.
You can rewrite it more compactly when you bracket expressions and realize that these are the derivatives of the connection ''Gamma''
$$\frac{\partial \Gamma_{\mu}}{\partial x^{\nu}} - \frac{\partial \Gamma_{\nu}}{\partial x^{\mu}} + \Gamma_{\nu}\Gamma_{\mu} - \Gamma_{\mu}\Gamma_{\nu}$$
You can only get the Riemann tensor by contracting the ''Ricci Tensor''. Notice that one alpha is on the upper indices and one is on the lower indices:
$$R_{\mu \nu} = R^{\alpha}_{\mu \alpha \nu}$$
Repeated indices means you automatically sum over these indices. The lowercase $$\mu \alpha$$ actually describe some rotation plain for a very small area displacement $$(dx^{\nu}, dx^{\mu})$$
You can also contract using the metric, for instance
$$R_{\lambda \sigma \mu \nu} = g_{\lambda \rho} R^{\rho}_{\sigma \mu \nu}$$
Can you guess which one is contracted? If you said $$\rho$$, you'd be right. What is $$g_{\mu \nu}$$ contracted with $$R^{\mu \nu}$$? It's just $$R$$ is the answer. You would get the curvature scalar by contracted the Ricci Tensor $$R^{\mu \nu}$$ and has this following form
$$\nabla_{\mu} R^{\mu \nu} = \frac{1}{2} g^{\mu \nu} \partial_{\mu} R$$
where we call $$\nabla_{\mu}$$ the covariant derivative. I think the covariant derivative originally came from work on fibre bundles. The property of a covariant derivative just has this form:
$$\nabla_{\mu}AB = A\nabla B + (\nabla A) B$$
The covariant derivative of $$g_{\mu \nu}$$ is actually zero.
$$\nabla_{\mu} R^{\mu \nu} = \frac{1}{2} \nabla_{\mu}g^{\mu \nu} R$$
$$\nabla [R^{\mu \nu} - \frac{1}{2}g^{\mu \nu} R ]= 0$$
This can be rewritten as a short-hand
$$R^{\mu \nu} - \frac{1}{2}g^{\mu \nu} R = G^{\mu \nu}$$
so
$$\nabla_{\mu}G^{\mu \nu} = 0$$
This is the local continuity equation for gravitational energy. As I said before, $$g^{\mu \nu}$$ derivative is zero, so what we have is
$$R - 2R = 0$$
and $$R=0$$ when there is no energy-momentum present.
So we learned the ''Einstein Tensor'' $$\nabla_{\mu} G^{\mu \nu}=0$$
The right hand side of $$\nabla_{\mu} R^{\mu \nu} = \frac{1}{2} \nabla_{\mu}g^{\mu \nu} R$$ describes the matter in a universe.
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