"The most beautiful thing we can experience is the mysterious" ~ Albert Einstein

Thursday, March 28, 2013

Columbine Peak (12,662')

Well, I didn't want to wait until the summer to blog about the Sierra, so I figured why not blog on one of our older, favorite hikes.  Columbine Peak!  (Well, that, and I'm kind of bored tonight ... ) This also completes all the topics/categories I plan to have on this blog.  Rebecca and I both really enjoyed this outing back in August, 2008.  It was also the hike where I first introduced Rebecca to the pre-dawn starts I'd recently started growing fond of.  The day prior we had climbed Mt Emerson and camped that night in Four Jeffrey's Campground.  We always like that campground and have stayed there several times now.  There are a few sites that are down in the aspens by Bishop Creek - choice spots!

We got up and started hiking by 3AM, most likely disturbing our fellow campers as we packed up and made our way to the trail head.  I love early morning starts.  I think it really helps for long hikes because later you're not sure if you dreamt the first part of the hike, or really did it.  Nice little psychological boost for later in the day, as it almost feels like you didn't hike as much as you really did.  Watching the day come alive is another bonus.  It feels like it pulls you out of a dark, dreamy world into a brand new day full of energy.  Our day just started to come alive 5 miles into the hike at Bishop Pass, as we peered over into Dusy Basin

Dusy Basin and The Black Divide From Bishop Pass, Daybreak
Our objective for the day - Columbine Peak - came into view for the first time near Bishop Pass, but the trip had a treat in store for us before that.  Our first visit to Dusy Basin - a high alpine basin on the far side of the pass.  Columbine Peak lay a couple miles on the far side and required a short trek across this pretty basin amidst alpine flowers and lakes, with distant views of the Black Divide on the western skyline above the deep chasm of Le Conte canyon, which dropped out of site over the western edge of the basin.  The Palisades towered over the eastern side of Dusy Basin looking dark and brooding in the early morning light.  There were some other souls in Dusy basin enjoying the majesty of the mountains that morning, however, I'm not sure they appreciated us stumbling upon them and disrupting their morning routine.

Deer in Dusy Basin
We dropped down into Dusy basin and slowly made our way across.  I was actually getting a little uncomfortable and nervous at this point, because my upper back was bothering me.  It was one of those feelings where it feels like it just might go out and put you on the ground.  Not the most exciting prospect in a remote basin many miles from the car, but it seemed to be holding together so we carefully pressed on.  The beautiful views of the alpine lakes in the basin beckoned us onward.

Giraud Peak Over Dusy Basin
Eventually, we got to the base of Isosceles Pass, from which our route up the NE Ridge of Columbine Peak rose above.  Getting to the pass was reported to be difficult, but it turned out to be not all that bad.  The big boulders were there, as reported, but everything was solid.  No loose stuff, just a nice boulder hop all the way to the pass.  From the pass we had nice views down the other side into Barret Lakes basin and the NE Ridge of Columbine rose above us.  It was time for a break and a small refueling before heading up the ridgeline.  The climbing was really enjoyable from here on.  Maybe an easy Class 3, with little exposure for the most part.  Just enjoyable scrambling for the next 600-800 feet.

Looking Up Towards The Summit of Columbine
The summit had incredible views.  The Black Divide stretched across the sky to the West, while the Palisades loomed high in the East.  The fluted, chiseled steep cliffs on the west side of the Palisades are truly impressive.  Then again, their eastern aspects are equally impressive.  Devils Crags rose up on the southern end of the Black Divide.  These are remote, ominous peaks known for their loose rock and wild surroundings.  We could see back to Bishop Pass from whence we came.  It looked rather far away now on the other side of Dusy Basin.  The uphill on the way back to Bishop Pass looked flat from this vantage point, but we had some heavy breathing in store for us later in the day when we had to huff it back over.  On the south side of the summit was the impressive summit diving board.  Not the kind of diving board one wants to jump off, though.

Columbine Peak Summit Diving Board
After an enjoyable stay on the summit, we made our way back down to Isosceles Pass and then down to Dusy Basin again.  We took our time going back across, visiting some other small alpine lakes.  The Black Divide looked gorgeous over the eastern end of Dusy basin.  It's obvious from the basin that a canyon lies between the two, but one would never guess it's a 3000+ foot drop down from Bishop Pass and a 4000+ foot rise up to the high points of the Black Divide on the far side.

Black Divide Above Dusy Basin

Rebecca also came across some rare Columbine flowers, which was rather fitting for the occasion!  These were a hybrid mix between the red and white columbines.  There were lots of other flowers around including plentiful amounts of rock fringe.

Rock Fringe
Rebecca and I decided we were due for a well-deserved rest break by this point and we made our way over to Lake 11,388 and soaked in the views of the Palisades from the lake's grassy shores, while we enjoyed some lunch.  My back was feeling better by now and it felt nice to lay down and relax.  The warm sun almost put me to sleep, but I always find it hard to actually fall asleep and take a nap like that, despite how relaxed one can feel in the warm sun after a long, hard hike.  Besides, if I actually fell asleep I'm not sure I would have had the motivation to do the uphill grind back to Bishop Pass in my groggy state.  We knew it was inevitable so we got up and crossed the outlet stream near the lake and made our way to the slopes below Bishop Pass.

Crossing Lake 11,338's Outlet Stream

It turned out to not be all that bad going back up to Bishop Pass.  It's only about 600 hundred feet of gain after all, but the pass was at 12,000 feet and we were tired by this point.  When we got there we enjoyed one last view of Dusy Basin and headed down the homeward side.  A pack train was also making its way down and it was tempting to jump on the back mule for a free ride down.  I'm not so sure the mule, or the pack-train driver, would have appreciated that, so we decided to just hike it.  This part of the hike was in the dark on the way up, so we made sure to enjoy the views by Long Lake on the way back, while we reveled in the day!

Long Lake, Homeward Bound

Saturday, March 23, 2013

Acai/Maqui Berry Bowl with Incan Warrior Energy Drink

One thing I have been enjoying about this new superfood trend Rebecca and I are on is all the new exotic foods to try, many of which I have never heard of before, like the Sacha Inchi seeds and Maca Powder that I already talked about in earlier blogs. They're always from faraway places like the Himalayan region for Goji Berries, or some high-altitude region in South America for Maca and, so on. 

This blog will cover a breakfast with two exotic fruits I also hadn’t heard of before: the Acai Berry and the Maqui Berry, as well as a drink with a real energy-kick! 

Acai and Maqui Berry Bowl

The Acai (Pronounced like Ahhhh-Sighhh-eeeeee) and Maqui berry both have a somewhat similar appearance to the blueberry and I think if you enjoy those, you’re going to love Acai and Maqui Berries. Although, they do both have their own characteristic flavor.

Both of these berries are antioxidant powerhouses, with a one-two knockout punch for any free radicals that dare cross paths with them.  Blueberries, which everybody knows are high in antioxidants, come in with an ORAC (Oxygen Radical Absorbance Capacity) score of 7000 per 100g. Acai comes in with a score of around 42,000 and Maqui Berries, which are the highest antioxidant containing fruit in the world, comes in with a score of 92,000.  Before you get too excited though, the ORAC scale is measured in vitro and has been dropped by the USDA for “insufficient evidence of biological significance”. Regardless, I don’t think a nutrient dense fruit could be a bad thing, could it?

This breakfast bowl is a pretty light breakfast, which I can’t eat every day, despite now being one of my favorites. I need something with a bit more protein on a daily basis. But, it really is delicious. It’s almost has a fruity ice cream like appeal to it, but it is very low in sugar.

It's a very slight modification to the Acai Bowl in Superfood Kitchen by Julie Morris.

Here are the ingredients:

  • 1 frozen banana
  • 1 Tbsp Acai Powder
  • 1 Tbsp Maqui Powder
  • ¼ cup milk of choice (I used almond or hemp milk)
  • A bit of stevia for sweetness

Throw those guys into a food processor and mix.  Easy, Peasy!  I'll also sometimes throw some extra bananas and granola on top, along with some frozen blueberries.

I’ve also thrown in some Psyllium Husk for extra fiber and Maca powder for an energy kick, but I find the flavor is never quite as good with these guys included. Simple seems best, with this one.

I never thought I was the best photographer, but I guess I do okay in the outdoors.  Take 300 pictures per trip and you're bound to get something good, right?  But, this food photography is a bit more challenging.  For one, when the food is done, I just want to eat while the eat'n is good, ya know?  Then, the indoor lighting is always a challenge.  I also don't really have the patience for the setup to make the pictures look all nice, either.  I'm trying, though.

In the meantime, Rebecca helped me with these.  She's like a pro.  The first one with the Incan Warrior Energy Drink is my photo, which is clearly the least visually desirable .  The next two photos, which are much nicer, Rebecca took of another Acai-Maqui Berry bowl I made.  The last photo is of an Acai bowl (no Maqui here) Rebecca made a ways back.  Notice the color difference between the pure Acai bowl and the ones with Maqui Berry.  Maqui Berries have a deep purple pigment!

The recipe for the Incan Warrior Energy Drink is down below.

Acai and Maqui Berry Bowl with the Incan Warrior Energy Drink
Acai-Maqui Berry Bowl

Acai-Maqui Berry Bowl

Rebecca’s Acai Bowl

Incan Warrior Energy Drink

This drink would make a great coffee alternative. It sort of tastes like a dark chocolate latte, or something similar. Also, I think the Yerba Mate could be replaced with a caffeine-free alternative. I personally don’t do well with caffeine, so I plan on trying this again with Organic India’s Goto Kola Tulsi Tea, but I think any dark roasted, or earthy tea, would go well in this drink. The cacao and maca should provide plenty of energy without the caffeine, anyhow.

Also, for comparison with the fruits above, cacao comes in at 95,000 on the ORAC scale. Yikes! 


2 Tbsp Cacao
1 Tbsp Maca Powder
1 Tbsp Yacon, or Agave, Syrup
1 Scoop Amazing Grass Chocolate Powder, or something similar
1 Cup Milk of Choice (I used hemp milk)
1 Cup brewed Yerba Mate

Simply combine ingredients in a blender and have at it.  If you don't get an energy kick from this guy, something is wrong  ;-)

UPDATE:  I recently made this with 1 Cup Peppermint Tulsi tea and 1/4 tsp Peppermint Extract, in place of the Yerba Mate.  This a delicious caffeine free alternative!

This recipe came from Navitas Naturals: Incan Warrior Smoothie 

Ingredient Highlight - Maqui Berry!
The Maqui Berry is a staple of the Mapuche Indians in Chile.  Is it a coincidence that the Mapuche Indians are some of the longest-living people in the world?  Perhaps ;-)  As I mentioned above already, Maqui Berries are the highest known antioxidant rich fruit in the world, at 92,000 ORAC pts per 100g.  The Maqui Berry is particularly strong in two antioxidants of the flavonoid variety - polyphenols and anthocyanins.  Studies indicate these help repair and protect DNA.  Additionally, studies show that anthocyanins are beneficial when it comes to anti-aging effects, since it helps to reduce oxidative stress, as well as improve brain function.  Maqui Berries are also rich in Vitamin C, minerals and help to fight inflammation.   Did I mention they are delicious too!**

**Julie Morris, Superfood Kitchen

Cosmological Redshifts and Fun With Relativity

Thought I would start off my first physics blog with a topic I always thought was kind of neat – cosmological red shifts. Also, when I first learned about this, I realized how I was walking around with some mistaken notions for quite some time and how descriptions from popular physics, or popsci, promulgate these mistaken notions.

Before I start, I should warn this is a somewhat contentious issue and interpretation is important.  I'll try to present a simple description, which will probably oversimplify the matter too much, but at least make for an entertaining blog ... I hope!  I will leave links at the bottom to some professional physicists, where they duke out the matter in greater detail, for those interested.  At the very least, I think you'll see the whole business of redshifts in Cosmology isn't quite as simple as popsci can make it sound at times.

The complications arise between what we mean by a Doppler Shift and what we mean by a Cosmological Red Shift.  They are different.  But, first, let’s start with what a wave is.  The easiest way to get a feel for a wave is by imagining ones we are all familiar with – water waves. 

Water Waves

A water wave is the propagation of energy through water. As you can see, there are troughs and crests, which correspond to the wave amplitude. Also, we can see that the waves repeat. The length of one unique segment (crest-to-trough-to-next-crest) is the wavelength. The number of crests that pass by a fixed point in a fixed amount of time is called the frequency of the wave. There is an inverse correspondence between wavelength and frequency – high frequency means small wavelength and vice versa. We can represent all this with simple plots, like the following.

Sounds waves are the propagation of density perturbations in air. Electromagnetic waves are the propagation of electric and magnetic fields. Light is one kind of electromagnetic (EM) wave, or specifically a certain frequency/wavelength range of the EM spectrum.  Most waves require a medium to travel through (like the water, or air) but electromagnetic waves can travel through empty space. This is due to how the electric and magnetic fields interact with each other – a time varying electric field creates a magnetic field, and vice versa, which is very evident when looking at Maxwell’s Equations.

Now, when a source like the horn on a train is moving, the waves will get compressed and stretched (i.e. their wavelength will change). In the direction of motion the wave crests will get compressed and in the opposite direction they will get stretched.  Everybody is familiar with this by how the pitch of a train is very high when it comes toward you, but then decreases after it passes. This image visually depicts the waves beings stretched and compressed.

With light, wavelength corresponds to color. Blue-ish light is higher frequency, smaller wavelength. Red-ish light is lower frequency, larger wavelength. We, therefore, say the light is red-shifted or blue-shifted by a moving source. 

All this is the familiar Doppler Shift. It comes into play in astrophysics quite often. Stars moving away from us have their light red-shifted. Stars moving towards us have their light blue-shifted. On a larger scale, galaxies moving away from us have their light red-shifted. Galaxies moving towards us have their light blue-shifted.

We may all be familiar with this from the Big Bang, which says the Universe has expanded from a small, dense region a finite time in the past. Spacetime itself was created in the Big Bang, so it is spacetime itself that is expanding. The Big Bang is actually a pun on words, because there was no “bang” that happened in some pre-existing space. In addition, our physics are only valid back to around 10^-44 seconds (called the Planck Time) and therefore the Big Bang says nothing about the actual “bang”, or moment of “creation”.

So, as spacetime expands, it is taking all the galaxies along with it, similar to how flowing water sweeps things along with it. The way it works is compared to an inflating balloon. Using the illustration below, we can see that all points on the balloon will view themselves as if they are at the center of expansion, just as we do on Earth.

Therefore, we should see the light from distance galaxies predominantly redshifted. The further a galaxy is away from us the faster we will see it receding away and the more its light will be redshifted. Here’s where we get into trouble. If we think about this redshift as solely a Doppler Shift, we find that the some very distant galaxies are receding away from us at faster than the speed of light!   But, we know from Special Relativity that nothing can travel through space faster than c (i.e. c = the speed of light).

What’s going on here?

I’ll give a hint – cosmological redshift is part of the answer!

Let’s also be more specific about what Special Relativity (SR) says. It says nothing can move through space faster than c, or the speed of light.  It doesn’t say anything about the expansion rate of space itself.  So, even though a galaxy cannot move through space faster than c, it can be swept away by the expansion of space at faster than c.  In addition, SR says anything with mass must travel slower than c, but massless particles, like the photon (i.e. light), must always travel at c (through a vacuum), as viewed by any observer.

A cosmological redshift is caused by the expansion of spacetime itself. In other words, a galaxy can appear to have red-shifted light even if it’s not moving through space. This effect can become predominant on cosmological (or, vast!) distance scales.

In addition, in a curved spacetime (or, expanding spacetime) we cannot even clearly talk about relative velocities between two objects and, therefore, cannot clearly define a Doppler Shift.   Let’s see why.

Normally, when we add two vectors A and B in a flat space, we’ll move the tail of B to the tip of A, then we’ll draw a new vector that goes from the tail of A to the tip of B and this vector is then (A + B). Note that when we move B we keep it parallel – this is called parallel transport. We can also use the parallelogram technique, which is illustrated in the next figure.  Notice that the dashed line representing the top perimeter of the parallelogram is parallel to and has the same length as A, likewise with the other dashed line and B.

Vector Addition

The problem arises when we see that parallel transport through a curved space does not necessarily leave you with the same vector you started out with. To make matters worse, parallel transport is path-dependent. Depending on what path you drag B along to get to the tip of A, you may end up with a different B  and, therefore, a different answer for A + B. This can be seen in the following images. As a vector gets parallel transported around a curved surface of a sphere, we end up with the vector pointing in a different direction once it is returned back to its original position.  Note the difference between the red and blue arrows below.

Parallel Transport of a Vector on a Curved Surface

The more dynamic (i.e. curved, expanding, etc.) the space the more convoluted this becomes. Now, if vectors A and B represent the velocities of two galaxies labeled A and B, and the space is highly dynamic between them, we can see how defining the relative velocity between them (A + (-B)) becomes difficult.  In fact, to say a galaxy is receding away from us, which implies relative velocity, is a mathematically imprecise statement in a dynamic, curved spacetime.  It's one reason why we find things receding away from us faster than the speed of light, if we think of all this solely in terms of Doppler Shifts and galaxies moving through space.

Let’s now look at two scenarios, or thought experiments, that will illustrate further the difference between a Doppler Shift and a Cosmological Redshift. Then, for those of you who haven’t fallen asleep yet and are eager to see more details, we’ll do a small mathematical derivation that will more deeply illustrate what a Cosmological Redshift is!

For the first scenario, let's take two galaxies, labeled Galaxy A and Galaxy B, that are at rest with respect to each other, as depicted in step (i) in the image below, and have galaxy A emit a photon in the direction of Galaxy B.  In step (ii), let's move the galaxies through space and bring them back to rest before the photon reaches galaxy B.  (Remember, this is a thought experiment, so we can pretend to be God and hurl things around the Universe at will!)  Now, when the photon reaches Galaxy B in step (iii), there will be no Doppler shift, because the galaxies are at rest during emission and absorption.  Doppler shift depends on relative velocities.

No Doppler Shift (or any shift) since galaxies are at rest with respect to each other at emission and absorption

The next scenario is very similar, except for step (ii).  This time, in step (ii), we are NOT going to move the galaxies through space.  Instead, we are going to expand the space between the galaxies, thereby effectively making them the same distance apart that they achieved in the previous scenario.

In this scenario, the photon experiences a redshift!

This time, when Galaxy B detects the photon, a redshift is found!  Despite the fact that steps (i) and (ii) are identical and despite the fact that all the distances are the same, the photon is redshifted in this scenario and this is solely due to the expansion of space.  An expanding spacetime redshifts photons, or causes them to lose energy.

Okay, still awake?  How could you not be awake and riveted at the edge of your seat at this point ;-)

Let's push on and do some math that will show this result more explicitly!  What I hope to achieve in this section is to give the reader, who has a basic understanding of calculus and tensor notation, a feel for some of the concepts in Special and General Relativity and hopefully a greater appreciation for the topic of this thread.  For those without the math background, there may still be some neat gems from Relativity waiting for you.  Just gloss over the equations.

We'll start with what's called a simplified Robertson-Walker metric.  (The full Robertson-Walker metric characterizes our actual Universe, but it is much more simple and heuristic to use this simplified one, as the end result is the same in both cases)

To get a little more familiar with the metric, think of the Pythagorean Theorem for calculating distances (x² + y² = z²).  Notice how similar this is to the Robertson-Walker metric.  This is all a metric does - it calculates distances.  However, through Special Relativity, we have learned to treat space and time on the same footing.  Einstein unified the two and so now we are talking about distances not just in space, but rather distances in spacetime!   This also means we are now dealing with 4 dimensions (3 spatial, 1 time) and not the usual 3 spatial dimensions.  a(t) is called the scale factor and, as you can see, it is dependent on time and has an effect on the spatial dimensions.

The little minus sign on the time part above, characterizes the signature of the metric.  This minus sign is partially responsible for some of the cool features we find in Special Relativity.  This is called a Lorentzian signature.  If there was a plus sign there instead, we would have a 4-dimensional Euclidean metric and, other than the extra dimension, this isn't nearly as interesting!  (As a side note, it is possible to transform between Lorentzian and Euclidean metrics using what's called a Wick Rotation.  This is done all the time in Quantum Field Theory calculations to simplify the math)

We will now use a variational approach to calculate what are called the Christoffel Symbols (more on those below) and obtain the geodesic equation (more on that below, too) for this simplified Roberston-Walker space.  We're basically going to be solving for curves of maximum proper time, which correspond to the shortest-path through a curved space, which is basically what a geodesic is.  This is all somewhat similar to deriving the Euler-Lagrange equations of motion by varying the action and using Hamilton's Principle of Least Action.  Anyhow, we won't derive it here, but the following formula is used, which utilizes proper time, τ, for it's parameter. 

Plugging in for our specific metric, gives:

We'll drop the y and z spatial dimensions for simplicity and have used the Kronecker delta to keep things clean.

Now, let's first vary with respect to time, meaning:


We neglected terms that are second order, or higher, in the infinitesimal variation and we also skipped a step or two on the second line.  Plugging these into our variational formula and only keeping terms that contribute to first-order changes in I, we get:

We want to solve for vanishing δI, or vanishing first-order changes in I, because this is indicative of a critical point, or point of maximum proper time, as discussed above.  In order to get a reasonable, and useful, expression for when δI vanishes from the equation above, we need to get that δt out of the derivative in the first term, so that we can pull it outside of the square brackets.  We can achieve this by integrating the first term by parts, ultimately giving us:

In order to make this vanish for any arbitrary δt, the expression in parentheses must vanish.  This is then equivalent to the geodesic equation,

from which we can read off the Christoffel Symbols.  We don't actually need to obtain all the Christoffel symbols for the problem we're tackling, but because they're interesting let's talk about them anyhow.  We'll also talk about the geodesic equation, which we do need for the problem at hand.

Christoffel symbols are also called connection coefficients.  Put very simply, they provide a connection between vectors that lie close to each other in a curved space.  Put more technically, they provide a connection between vectors in nearby tangent spaces.  The covariant derivative quantifies the instantaneous rate of change of a vector (and more generally, a tensor field) in comparison to what the vector (or, tensor)  would be if it were parallel transported.  The Christoffel symbols provide the connection to transport vectors from one tangent space to another and are a crucial part of the covariant derivative.  All of this would be a moot point if we were in a flat space.  In the limit of vanishing curvature, the Christoffel symbols also vanish and the covariant derivative becomes the regular derivative we're all familiar with in flat space.

The geodesic equation is the generalization of a straight line in curved space.  In other words, a geodesic is the shortest path in curved space.  If you take a look at the geodesic equation above, you'll see that when the second term vanishes as a result of the Christoffel symbols being set to zero, we are left with the equation of a straight line in flat space.

Moving on, let's set the term in parentheses, which is essentially the geodesic equation, to zero, like we said we would above and use the affine parameter, λ, instead of the proper time, τ.  An affine parameter is any one that can be related to the proper time in a linear fashion (λ = a* τ + b).

This is a differential equation, which is solved by the following:

We'll use this below.

Now that we're using λ, we can take a look at a null path. Null paths are paths of zero spacetime distance and they are the paths followed by massless particles, such as photons.  Let's think about what this means. First, ds² (or ds) is called the interval and it's an invariant.  Although space and time are now relative, every observer will always agree on the interval between any two events.  Also, recall that space and time are on the same footing in Special Relativity.  This means we must now think of motion, not just through space, but through time, as well.  Sitting at your monitor reading this blog, all your motion is in the time dimension with zero motion in the spatial directions.  Looking at the Robertson-Walker metric and setting the scale factor a(t) equal to 1 (which reduces it to the flat-space metric of Special Relativity), we notice a null path means that a photon's motion is split evenly between motion through time and motion through space.  Since the interval is invariant, once you start moving through space, you take away from your motion in time.  This is why you see the clock of somebody moving with respect to yourself ticking at a slower rate than yours.  (This is, of course, the logic that also clears up the famous Twin Paradox)  Notice that motion through space faster than motion through time is not allowed since this would make the root of ds² negative.  There's the Lorentzian signature at work for you.

Also, since the interval is fixed between any two events, this means everything moves through spacetime at the same rate - the speed of light.  It's just a matter of how your motion is shared between time and space.  A photon, or any massless particle, maximizes its motion through space, such that it is equal to its motion through time and, therefore, always travels along null paths.  Motion through space greater than this are not allowed.  Nothing can travel faster than c through space, but the combined motion through space and through time of any object is exactly the speed of light.  We humans have most of our motion through time and therefore move through space very slowly compared to a photon. 

This is also why we now switch to the affine parameter, λ.  A photon has zero proper time, (τ = 0), since proper time vanishes along null paths.  Proper time is the rate time ticks when you are at rest (ds² = -dt² = -dτ ²).  A photon can find no rest, since it is always stuck moving at the speed of light, c.  One can never find a reference frame, in which a photon is found at rest.  Every observer, regardless of their motion, always reports finding photons to be traveling exactly at c. This essentially means a photon does not sense the passage of time.  This is how we now know neutrinos have mass.  Once they realized neutrinos could undergo what are called neutrino oscillations, which is a time-dependent phenomenon in the frame of the neutrino, this meant neutrinos could not travel at c and must have mass.  This helped to solve the solar neutrino problem.

The equation for a null path in our simplified Robertson-Walker metric gives us:

Next, lets go to a frame that is comoving with the photon in question.  This frame has 4-velocity:

You might be wondering why this vector is 4-dimensioanl.  This ties back to special relativity uniting space and time.  The first component is referred to as the "time-component", while the remaining three are the usual spatial components.  So, space and time are united not just conceptually in SR, but even more fundamentally at the mathematical level.

Although we won't show this here, 4-velocity is always normalized by

Let's pause one more time to see what this means.  This normalization sets the magnitude of the 4-velocity.  Since this is a tensor equation, if it is true in one frame, it is true in every frame.  This means the 4-velocity is always fixed, which amounts to the same thing we said above.  Everything moves through spacetime at the same rate.  It's just a matter of how much our movement is shared between space and time, but the magnitude is always fixed.

Now, let's solve for the energy of the photon.

Next we can consider the energy of the photon emitted at scale factor a1 and time t1, and absorbed at scale factor a2 and time t2, and divide the two energies.  This is exactly the second scenario we looked at above.

This amounts to the following redshift

 And, there you have it!

This is a Cosmological redshift.  Nowhere in this formula are relative velocities mentioned, which we would need for a Doppler Shift.  Only the scale factors, which control the expansion of space, are referenced and control the redshift!

Wasn't that like totally cool?  I hope, at the very least, I have provided you with plenty of ice-breakers for starting up conversations at the next party you attend.  Trust me, you will be the coolest guy/gal around, if you do.   ;-)

Here are the links to the professionals I promised.  Reading through those will provide additional insight, but I think you'll find agreement isn't 100%.  At the very least, I think you'll find this whole business of redshifts isn't as simple as it's usually made to sound in popsci.

Sean Carol's Blog
Ted Bunn's Blog I
Ted Bunn's Blog II

Well, now that I'm wrapping this blog up, I gotta admit, making all those equations was a pain in the ass at first.  I think I might be getting the hang of using the LateX equation editor, which generates the images of these equations.  Regardless, it is pretty obvious these physics blogs are much more work, if they're going to include math.  I'll still write them up from time to time, but they probably won't be as prevalent as some of the other topics.

Have a good one!

Wednesday, March 20, 2013

The Lean, Mean, Green Smoothie

I keep hearing about how healthy green juices and green smoothies are supposed to be, so I figured I should try a few. Also, Rebecca has been making a yummy green smoothie for breakfast that I tried on several occasions. I recently came across one called “Workout Recovery Smoothie” on the Navitas Naturals web page, so I started making that one not too long ago for myself.   I even used it to recover after a workout, or two - imagine that!

Since then, I’ve modified some of the ingredient amounts and added in quite a few too. So, maybe I earned the right to rename it and call it my own recipe? Well, probably not, but … I call it the Lean, Mean, Green Smoothie, and it’s quite possibly the greenest smoothie on the planet.

Here’s the impressive lineup of ingredients. Kitchen counter space can be one of the main challenges to making this one.  Believe it, or not, an ingredient (Hemp Protein Powder) was missing because we were all out: 

First thing you gotta do is start soaking the Goji berries, unless you actually have fresh ones, but most folks typically get them dehydrated.  You can see mine soaking in the image above, with the big jar of dehydrated berries right behind.

Add a medium-size handful of Goji berries to 1 2/3 cup of coconut water. Let them soak for about 10-15 minutes, or until they start to soften. See if you can solve the mystery as to why some float and why some sink. 

While that’s going on, you can start getting the rest of the ingredients out, as follows:

  • 1 frozen banana (sliced into small slices – before freezing!)
  • ¼ cup, or less, of Walnuts (too many will give it an unpleasantly smooth consistency. Trust me! I learned that the hard way!)
  • 1 Tbsp Hemp Protein Powder
  • 2 Tbsp Hemp Seeds
  • 1 tsp Chia Seeds
  • 1 Tbsp Maca Powder
  • 1 Tbsp Organic India Psyllium Husk (6 grams of fiber, right there)
  • 1 scoop Amazing Grass Chocolate (or Green) Super Food Powder
  • 1 tsp Wheatgrass Powder
  • 2 tsp Spirulina (or 1 tsp, if you don’t like the taste of Spirulina as much as I do)
  • 1 tsp Noni Fruit Powder
  • 1 handful Kale
  • 1 handful Spinach
  • Stevia for sweetness, although I find it usually doesn’t need it. 

(Stevia is a great sweetener because it’s a herb and doesn’t have the negative health consequences sugar does. The taste doesn’t agree with some folks, though.)

Note: make sure the Kale is organic! It’s one of the most nutrient dense green leafy guys you can eat, but it’s also on the Dirty Dozen list, which means non-organic grocery-store-bought Kale is also likely to be covered in the most pesticides.

Once the Goji Berries are good to go, combine everything in a blender and go. 

Here’s all the ingredients in our lean, mean, green (errr, red) smoothie machine … a.k.a, a cheap-ass Black And Decker, but, hey, it gets the job done (most of the time …)

Who’s got a Vitamix they want to trade with us?

Add in more coconut water, or even plain water if you want to thin it out a bit. Ice cubes are another option if you want it a bit “chilled”. Yet another option is to use Water Kefir instead of the coconut water, making this a potent probiotic drink too!

Anyhow, you should end up with a very green-looking concoction like the one below. Doesn't it look yummy? Hard to say sometimes, but it sure tastes yummy.

The Lean, Mean, Green Smoothie

Doesn’t get any greener than that!

Would you like some smoothie in your chlorophyll?

If you’re new to green smoothies, I’d probably recommend starting with the Navitas Naturals recipe and see how you like that one first. It’s really good, too. As a matter of fact, Rebecca prefers that one over this recipe.


Ingredient Highlight: Spirulina!
Spirulina is a blue-green algae that has been living on the planet since the appearance of life.  The green color is derived from chlorophyll. The blue color is derived from the potent phytopigment phycocyanin (I have no idea how to pronounce that either!).  Phycocyanin appears only in Spirulina.  Being one of the simplest and earliest lifeforms it has a long history of developing and sustaining the food chain, providing the fundamental nutrient and food sources for life.  Spirulina is 65-71% protein by weight.  And, it is a complete protein source, containing all eight essential amino acids, and eighteen amino acids in total.  (An essential amino acid is one your body cannot make and can only get from diet)  Is rich in vitamins A, B1, B2, B6, E and K.  It also contains many phytonutrients, caretenoids, Essential Fatty Acids (EFAs) and minerals like Iron, Magnesium, Selenium and Zinc.  Research indicates that Spirulina helps support the immune system, helps support and maintain blood pressure, assists in heavy metal detoxification, helps support optimal gastrointestinal flora and more.  It's quite the superfood! **

Book Review and Parapsychology Primer

Sam Parnia's highly anticipated book, Erasing Death, was recently published .  The level of anticipation was due to the fact that Dr. Parnia is the lead researcher for the AWARE (AWAreness During REsuscitation) project.

Horizon Research Foundation - AWARE

AWARE is taking place in several hospitals, covering a few countries, and seeks to improve upon and gain more insight into resuscitation science.  However, there is a particular area many folks are interested in.  AWARE also seeks to objectively validate that the Out-Of-Body (OBE) experiences, which somewhere between 2-5% of cardiac arrest patients experience, are real.  In other words, AWARE is trying to validate that consciousness can exist independent of the body ... at least, for a limited amount of time.  This would be truly astonishing, if true.

Although, it is a must read for anybody interested in parapsychology, I didn't think it was a super riveting read.  Partly because AWARE is still in its early stages and also because it's not really meant to be a book on parapsychology.  It is meant to be about resuscitation science, as Sam Parnia is one of the top doctors in this area.  Of course, there is plenty of overlap between the two and the book does set down a few remarkable precedents for parapsychology.  Before, I get into that ... what is parapsychology?

When most people hear that word they think of images Hollywood has helped create: Ghost Busters, Close Encounters of the Third Kind, and perhaps something like the Exorcist.  So, let me clear things up with a simple definition often used for parapsychology.

Parapsychology is the scientific investigation into anomalous phenomena, typically related to consciousness.  Let's look at an example, that will help clear things up even more.

One of the bigger areas of parapsychology is called psi, which covers things like telepathy and precognition - things we normally associate with the phrase "psychic ability".  Once again, we normally think of things like the local palm reader, or some gypsies with a crystal ball.   In actuality, many psi experiments intentionally deal with average folks like you and me that have no professed psychic ability.  What one class of psi experiments show is a small effect indicating we all have access to information, which is gained in an extra-sensory fashion, albeit at an unconscious level.

Probably the most popular and perhaps the most conclusive experiment, or set of experiments, showing this effect is called the Ganzfeld.  Essentially, two people are separated in two different chambers that are electromagnetically shielded.  One is the receiver, one is the sender.  The receiver is in a dark reddish-light, with eyes covered and wearing headphones that apply white noise (i.e. they are sensory "starved").  The sender is given an image, which is part of a set of 4 visually disparate images, all randomly selected via computer from a large database of images. This is done in a double-blind fashion so nobody knows what the image will be beforehand, even the experimenters.  The sender mentally sends the image and the receiver is instructed to tell what impressions he/she is receiving.  Later, an image is chosen from the four visually disparate images that fits the impressions recorded.  If there was no effect and this was due to random chance, one would expect a hit rate of 25%.  The value being converged upon after performing a meta analysis of many replications across different labs is 32%, indicating more correct choices than random chance would predict. That doesn't sound like much, but it turns out to be a highly statistically significant result.

You've probably heard about the Higgs particle by now.  In fact, CERN just announced the Higg's has been "officially" discovered at the Large Hadron Collider (LHC).  But, this official announcement and discovery also depends in a large part on statistics.  The Higgs particle wasn't discovered by directly detecting it.  It decays way too fast for that.  But, the math of the Standard Model of physics predicts what the signature should look like from the decay of a Higgs particle.  The catch here is that the signature must be detected out of a lot of background noise.  More and more data is collected until the statistical analysis says that the chance of the result being a fluke from random chance is greater than 1 in 1 million, otherwise know as 5-sigma significance.  This is the standard level of statistical significance expected for an official scientific claim of a new discovery.  It will take much more data before they can even nail down the properties of this new Higgs particle and determine whether it is the standard-flavor Higgs predicted by the Standard Model, or something more exotic, potentially opening the way to new physics.

For comparison, the meta-analysis for the Ganzfeld indicates roughly a 6-sigma significance.  To be a bit more precise, the Storm (2010) meta-analysis indicates approximately 46,948,356 to 1 with outliers removed and 8,695,652,173 to 1 without outliers, that the 32% hit rate is due to random chance, which implies rather strongly something real is happening.  This is an order of magnitude stronger than the data for the Higgs, at the time of their original announcement.  There are other statistical arguments that have been raised such as publication bias in the meta-analysis, but these have all been thoroughly addressed and covered elsewhere.  The Ganzfeld is one of the most heavily critiqued experiments, yet it still stands.  In fact, almost all hardcore skeptics, such as Richard Wiseman, have admitted that various aspects of psi have been reasonably demonstrated to the usual scientific standards.  But, for now they all stand by the claim, "extraordinary claims require extraordinary evidence". 

There are two problems here:  (1) There is no mathematical theory for psi, like there is for the Higgs.  A  reasonable objection and a concern I share.  There are proposals, but they're all incomplete and leave much to be desired.  (2)  There is a prevalent taboo in mainstream science against all things psi.  It would seem because of this second problem it's hard to achieve this subjective level of "extraordinary evidence" folks are looking for, because it's a little too easy to keep shifting the goal posts further and further out.

Anyhow, here is a decent interview of Dean Radin, who does a much better job of explaining the Ganzfeld than I do.  Dean Radin is a senior scientist at the Institute for Noetic Sciences, founded by Apollo astronaut Edgar Mitchel.

Back to the book!

Another area of parapsychology I find fascinating is Near Death Experiences (NDEs).  This is mainly because I think it has the strongest chance of being objectively verified independent of statistical analysis and has the strongest potential to be paradigm shifting.

The NDE phenomenon is so prevalent even mainstream science acknowledges that it is happening at clearly definable rates in cardiac arrest patients - approximately 10% of patients have an NDE (or at least remember having one), and approximately 3% contain a veridical OBE element. Veridical means perception that seems to be independent of the body (i.e. the patients have actually reported observing their body and resuscitation efforts from the other side of the room and much more).  Overall, Parnia and others say the number of NDE reports is now in the millions.  Here is one somewhat famous case, that of Pam Reynolds, as it was covered on BBC.  (AWARE has the potential to knock the socks off this one, though)

Pam's experience is fairly typical of the NDE experience, which has universal themes found across all cultures, religions and age groups.  12 such themes have been identified and are listed here.  (After listening to NDE'ers talk about their experience, I would have to say these short descriptions probably do little justice in expressing the power of the experience)

1. Out-of-body experience (OBE): separation of consciousness from the physical body
2. Heightened senses
3. Intense and generally positive emotions or feelings
4. Passing into or through a tunnel
5. Encountering a mystical or brilliant light
6. Encountering other beings, either mystical beings or deceased relatives or friend
7. A sense of alteration of time and space
8. Life review
9. Encountering unworldly (“heavenly”) realm
10. Encountering or learning special knowledge
11. Encountering a boundary or barrier
12. Return to the body, either voluntary or involuntary

The NDE debate revolves around whether these experiences are brain-induced, or happening independent of the brain.  Most of the medical mainstream would say they are brain-induced, but there is currently no brain-based theory that is satisfactory for explaining NDEs.   (In fact, consciousness itself is mostly unexplained.)  Currently, the only real reason to insist on this explanation as the correct one is that it fits within our current materialistic worldview, which isn't a very scientific way of operating.  One should at least keep an open mind.

For the first time ever, AWARE is attempting to offer strong evidence obtained from a controlled scientific experiment, utilizing the veridical OBE component of the NDE, as to whether, or not, the NDE is brain-induced, happening independent of the brain, or possibly some combination of the two. This is the first thing remarkable about this book

(1)  Sam Parnia is one of the leaders in resuscitation science in the world and this book is part of an effort to reshape and improve the quality of medicine delivered in that field.  Parnia is adamant that exploring all facets of the resuscitation process, including the Near Death Experience, is paramount.  Before, a reported NDE would be shrugged off as hallucination, at best.  Now, it may start to get more serious scientific study.

So, how are they going to objectively test the veridical Out-of-Body experience?

They're installing visual cues within hospital rooms, which up to now have been pictures that are out of visual sensory range of the patients body, but hopefully visible from the perspective of the OBE.   If a patient reports an OBE and correctly identifies the picture, it would lend a high level of confidence that perception is happening independent of the brain, especially if they can time the event to a period during which the brain was flat lined and the heart was stopped.  The results won't be fully conclusive, as they will need replications with tighter controls, to ensure their is no information-leakage, etc.

(2)  Unfortunately, they have not had any hits yet (i.e. patients identifying the visual cue), but they did have two OBEs, one of which was rather remarkable and covered in the book.  Unfortunately, due to logistics, visual cues were not installed in either of the two rooms the patients were in.  Additionally, the standard location of the visual cues would have been outside the visual angles reported in these NDEs, even if they were installed.  They are working to correct all this.

Anyhow, one patient correctly recalled events that were not visible to him during the cardiac arrest.  Here is a decent summary taken from a forum I participate in.

"One [NDE] was verified with medical records. The man identified a doctor he did not even know was in the room and who he never set eyes on. There was an important time marker- he heard the electronic defibrillator advise the shock in the correct words used. Two shocks were advised, which means he was in V-fib for 2-3 minutes between shocks as the device checks for heart rhythm. He was totally lucid and never had any alteration of consciousness. He noticed that the doctor he didn't even know was in the room was chunky and bald with blue scrubs and hat, but only saw him from behind. He saw the nurse wore lighter blue scrubs and was tall. He said he saw it all plain as day despite having his head partitioned by a curtain while they fed the cath up his femoral artery through his groin."

(3)  Lastly, the book went into the details of resuscitation science for the first half, which was actually pretty interesting.  Parnia now says death needs to defined as a process, rather than a moment in time.  It used to be that if (1) the heart was stopped, (2) an EEG indicated a flat-lined brain, and (3) the pupils were dilated with no reflex, indicating an inactive brain stem, a person would be considered dead.  But, they now know that the cells within the body, although not executing their normal functions, are still viable for hours after this point (some cells are viable longer than others).  In fact, patients haven been brought back several hours after they were "dead" (going by the old way that term was used) thanks to advances in resuscitation science.  Experiences from several of these patients, with at least one reporting an NDE, are covered in the book.

(4)  One last point that was interesting.  Parnia started out as a skeptic in all of this, or at the very least, as a professional fence-sitter who chose his words wisely.  Around the time of this book, his language has changed considerably.  Based on the evidence gathered so far,  he now believes something very interesting is going on and is leaning towards the view consciousness may very well survive bodily death.  He also thoroughly argues that all current conventional explanations for NDEs (oxygen deprivation, Ketamaine, or other administered drugs, a dimethyltryptamine (DMT) dump by the pineal gland, etc.) are unsatisfactory, although this was already pretty well known and accepted within the parapsychology community.

Here's an NPR interview with Dr. Sam Parnia that digs into much more details than I covered here, for those interested in more:

If AWARE does indeed record hits, which some feel is just a matter of time now, and this becomes part of mainstream science via the help of guys like Parnia, this will be truly paradigm shifting.

I'll wrap it up there in the interest of length and leave you all with a couple more videos on psi, for your enjoyment.  As you will see in these videos, the Ganzfeld is not the only experiment providing evidence for psi

Enjoy! ;-)

Saturday, March 16, 2013

Sacha Inchi Buckeyes

The only thing better than eating this delicious superfood treat is its name.  Sacha Inchi!!  It's a very rich and sweet dessert.  It's the kind of treat where you need to just eat one, muster up all the will power you can, and walk away.  Any more than that and the superfood benefits would probably get outweighed by the sugar intake.

This was our first time ever trying Sacha Inchi seeds and they have a very characteristic flavor that I can't describe.  Rebecca tried and ended up saying it was a cross between like 5 different things and I had no idea what she was talking about.  The flavor changes and builds up while you're chewing on them, and even more so in the form of this dessert.

The recipe is from Julie Morris' book called Superfood Kitchen.  Well, we have an older version called Superfood Cuisine, but I'm sure this is in the new edition too.

First, you gotta make the chocolate:

  • ½ cup solid cacao butter, chopped into shavings
  • 5 tablespoons cacao powder
  • 2 tablespoons agave nectar

There is no sugar in cacao, so the agave nector adds some sweetness.  You use a double-boil method to melt the cacao butter and then whisk in the cacao powder and agave nectar.  You have to keep it liquified for the next part.

  •  ½ cup Sacha Inchi Seeds
  • 6 tablespoons Coconut Palm Sugar
  • ⅔ cup soft Medjool Dates (about 6-7)
  • 4 tablespoons almond butter
  • 2 tablespoons Lacuma powder
  •  ½ teaspoon vanilla extract

You mix those ingredients in a food processor, adding water as necessary to get a sticky dough.  Add only a tiny bit of water at a time, or you will end up with a ball of cement.  Trust me!  I learned that one the hard way.  Next, roll the dough into tiny balls.  After those chill in the freezer for 20-30 minutes, take them out and dip them in the chocolate.  Giving you a few treats looking like the following:

Ingredient Highlight:  Sacha Inchi!
Sacha Inchi is a seed, despite it's large size and nutty appearance.  It has been enjoyed since pre-Incan eras, from which ceramics have been recovered with evidence of Sacha Inchi.  What I didn't know is that Sacha Inchi seeds are the highest known source of healthy plant-based Omega-3s.  Surprisingly, it has more Omega-3s than salmon - 13 times more, ounce for ounce.  Although containing less than Cacao, it's a decent source of the good-mood amino acid Tryptophan, making for a potentially mood altering treat when the two are combined! It's also rich in vitamins and minerals, including Iodine, which seems to be getting some attention, as of lately. **


** Julie Morris, Superfood Kitchen